#  $C^{m}$ Semialgebraic Sections Over the Plane

Charles Fefferman, Garving K. Luli

##  1 Introduction

In this paper we settle the two-dimensional case of a conjecture involving
unknown semialgebraic functions with specified smoothness.

Recall that a _semialgebraic set_ $E\subset\mathbb{R}^{n}$ is a union of
finitely many sets of the form

$\\{x\in\mathbb{R}^{n}:P_{1}(x),P_{2}(x),\cdots,P_{r}(x)>0,\text{ and
}Q_{1}(x)=Q_{2}(x)=\cdots=Q_{s}(x)=0\\}$

for polynomials $P_{1},\cdots,P_{r},Q_{1},\cdots,Q_{s}$ on $\mathbb{R}^{n}$.
(We allow the cases $r=0$ or $s=0$.)

A _semialgebraic function_ $\phi:E\rightarrow\mathbb{R}^{D}$ is a function
whose graph $\\{(x,\phi(x)):x\in E\\}$ is a semialgebraic set.

We define smoothness in terms of $C^{m}$ and $C^{m}_{loc}$. Here,
$C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of all
$\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$ whose derivatives up to
order $m$ are continuous and bounded on $\mathbb{R}^{n}$.
$C_{loc}^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ denotes the space of
$\mathbb{R}^{D}$-valued functions on $\mathbb{R}^{n}$ with continuous
derivatives up to order $m$. If $D=1$, we write
$C^{m}\left(\mathbb{R}^{n}\right)$ and
$C_{loc}^{m}\left(\mathbb{R}^{n}\right)$ in place of
$C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ and
$C_{loc}^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$, respectively.

To motivate our conjecture, we pose the following problems.

######  Problem 1 (Semialgebraic Whitney Problem; see [43].)

Fix $m\geq 0$. Let $\phi:E\rightarrow\mathbb{R}$ be semialgebraic. Suppose
$\phi$ extends to a $C^{m}_{loc}$ function on $\mathbb{R}^{n}$. Does it
necessarily extend to a $C^{m}_{loc}$ semialgebraic function on
$\mathbb{R}^{n}$?

######  Problem 2 (Linear Equations)

Fix $m\geq 0.$ Consider the linear equation

(1) $A_{1}F_{1}+\cdots+A_{D}F_{D}=f$

for unknowns $F_{1},\cdots,F_{D}$ on $\mathbb{R}^{n}$, where
$A_{1},\cdots,A_{D}$, $f$ are given semialgebraic functions. If equation
$\left(\ref{equation}\right)$ admits a $C^{m}_{loc}$ solution
$F_{1},\cdots,F_{D}$, does it necessarily admit a $C^{m}_{loc}$ semialgebraic
solution?

More generally, in place of (1) we can consider underdetermined systems of
linear equations.

Problem 1 was raised by Bierstone and Milman in [43].

Note that $m$ is fixed in the above problems so we are not allowed to lose
derivatives.

Problems 1 and 2 are instances of a more general question. The purpose of this
paper is to settle that question, and in particular provide affirmative
answers to Problems 1 and 2, in the case of
$C^{m}_{loc}\left(\mathbb{R}^{2}\right)$.

To pose our more general question, we set up notations and give a few basic
definitions.

Fix $m\geq 0$. If $F\in C^{m}_{loc}(\mathbb{R}^{n})$ and $x\in\mathbb{R}^{n}$,
we write $J_{x}(F)$ (the “jet” of $F$ at $x$) to denote the $m$-th degree
Taylor polynomial of $F$ at $x$.

Thus, $J_{x}(F)$ belongs to $\mathcal{P}$, the vector space of all such
polynomials.

For $x\in\mathbb{R}^{n}$, $P,Q\in\mathcal{P}$, we define
$P\odot_{x}Q=J_{x}(PQ)$. The multiplication $\odot_{x}$ makes $\mathcal{P}$
into a ring, denoted by $\mathcal{R}_{x}$, the “ring of $m$-jets at $x$”. We
have $J_{x}\left(FG\right)=J_{x}\left(F\right)\odot_{x}J_{x}\left(G\right)$
for $F,G\in C^{m}_{loc}\left(\mathbb{R}^{n}\right)$.

We consider vector-valued functions
$F=\left(F_{1},\cdots,F_{D}\right):\mathbb{R}^{n}\rightarrow\mathbb{R}^{D}$,
and we write $F\in C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ if
each $F_{i}\in C^{m}_{loc}\left(\mathbb{R}^{n}\right)$. We define
$J_{x}F=\left(J_{x}F_{1},\cdots,J_{x}F_{D}\right)\in\mathcal{P\oplus\cdots\oplus
P}$. Under the natural multiplication

$Q\odot_{x}\left(P_{1},\cdots,P_{D}\right):=\left(Q\odot_{x}P_{1},\cdots,Q\odot_{x}P_{D}\right)\text{,}$

the vector space $\mathcal{P\oplus\cdots\oplus P}$ becomes an
$\mathcal{R}_{x}$ module, which we denote by $\mathcal{R}_{x}^{D}$.

We will discuss $\mathcal{R}_{x}$-submodules of $\mathcal{R}_{x}^{D}$; we
allow both $\left\\{0\right\\}$ and $\mathcal{R}_{x}^{D}$ as submodules of
$\mathcal{R}_{x}^{D}$.

Fix $m,n,D$, and a subset $E\subset\mathbb{R}^{n}$. For each $x\in E$, let

$H\left(x\right)=f\left(x\right)+I\left(x\right)\subset\mathcal{R}_{x}^{D}$

be given, where $f\left(x\right)\in\mathcal{R}_{x}^{D}$ and
$I\left(x\right)\subset\mathcal{R}_{x}^{D}$ is an $\mathcal{R}_{x}$-submodule.
Then the family

(2) $\mathcal{H}=(H(x))_{x\in E}$

is called a “bundle” over $E$. $H(x)$ is called the fiber of $\mathcal{H}$ at
$x$.

###### Remark 1.1

We remark that our notion of bundle differs from the notion of a bundle
considered previously (e.g, [28]). In the present version, we do not require
$E$ to be compact and we require all the fibers $H\left(x\right)$ to be non-
empty.

When $m,n,D$ are not clear from context, we speak of a “bundle with respect to
$C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$”.

If $\mathcal{H}$ is given by (2) and $E^{\prime}\subset E$, then we write
$\left.\mathcal{H}\right|_{E^{\prime}}$ to denote the bundle
$\left(H\left(x\right)\right)_{x\in E^{\prime}}$, and refer to
$\mathcal{H}|_{E^{\prime}}$ as the restriction of $\mathcal{H}$ to
$E^{\prime}$.

A “section” of the bundle $\mathcal{H}$ in (2) is a vector-valued function
$F\in C^{m}_{loc}(\mathbb{R}^{n},\mathbb{R}^{D})$ such that $J_{x}F\in H(x)$
for all $x\in E$.

Note that sections $F$ belong to
$C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ by definition.

The bundle (2) is called “semialgebraic” if

$\left\\{\left(x,P_{1},\cdots,P_{D}\right):\mathbb{R}^{n}\oplus\mathcal{P\oplus\cdots\oplus\mathcal{P}}:x\in
E,\left(P_{1},\cdots,P_{D}\right)\in H\left(x\right)\right\\}$

is a semialgebraic set.

We can now state our general problem.

###### Problem 3

Let $\mathcal{H}=(H(x))_{x\in E}$ be a semialgebraic bundle with respect to
$C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. If $\mathcal{H}$ has
a section, does it necessarily have a semialgebraic section?

Again, we note that sections of $\mathcal{H}$ must belong to $C^{m}_{loc}$ for
fixed $m$, so we are not allowed to lose derivatives.

One checks easily that Problems 1 and 2 are instances of Problem 3.

Indeed, suppose $\phi:E\rightarrow\mathbb{R}$ is semialgebraic, as in Problem
1. Set $\mathcal{H=}\left(H\left(x\right)\right)_{x\in E}$, where

$H\left(x\right)=\left\\{P\in\mathcal{P}:P\left(x\right)=\phi\left(x\right)\right\\}\text{.}$

Then $\mathcal{H}$ is a semialgebraic bundle, and a section of $\mathcal{H}$
is precisely a function $F\in C^{m}_{loc}\left(\mathbb{R}^{n}\right)$ such
that $F=\phi$ on $E$.

Similarly, given an equation (1) as in Problem 2, set
$\mathcal{H=}\left(H\left(x\right)\right)_{x\in\mathbb{R}^{n}}$ with

$H\left(x\right)=\left\\{\left(P_{1},\cdots,P_{D}\right)\in\mathcal{P}^{D}:A_{1}\left(x\right)P_{1}\left(x\right)+\cdots+A_{D}\left(x\right)P_{D}\left(x\right)=f\left(x\right)\right\\}\text{.}$

Then $\mathcal{H}$ is a semialgebraic bundle, and a section of $\mathcal{H}$
is precisely a solution $F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ of equation (1).

In this paper, we settle the two-dimensional case of Problem 3.

###### Theorem 1

Let $\mathcal{H}$ be a semialgebraic bundle with respect to
$C^{m}_{loc}\left(\mathbb{R}^{2},\mathbb{R}^{D}\right).$ If $\mathcal{H}$ has
a section, then it has a semialgebraic section.

We give a quick sketch of the proof of Theorem 1.

By a change of coordinates and a partition of unity, we may localize the
problem to a small thin wedge

$\Gamma(c)=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}\in\left[0,c\right],0\leq
x_{2}\leq x_{1}\\}.$

More precisely, it is enough to prove that $\mathcal{H}|_{\Gamma(c^{\prime})}$
has a section for sufficiently small $c^{\prime}$.

We may assume also that our bundle
$\mathcal{H=}\left(H\left(x_{1},x_{2}\right)\right)_{\left(x_{1},x_{2}\right)\in\Gamma\left(c\right)}$
satisfies $H\left(\left(0,0\right)\right)=\left\\{0\right\\}$.

We analyze what it means for a given $F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$ with $J_{(0,0)}F=0$ to
be a section of $\mathcal{H}$. Our analysis produces finitely many
semialgebraic curves $\gamma_{1},\gamma_{2},\cdots,\gamma_{s_{\max}}$ in
$\Gamma\left(c\right)$, and we find that $F$ is a section of $\mathcal{H}$ if
and only if

  * •

$F\left(x_{1},x_{2}\right)$ and its $x_{2}$-derivatives up to order $m$
satisfy finitely many linear equations on the $\gamma_{s}$ and

  * •

$F$ satisfies finitely many linear equations on
$\Gamma(c)\setminus\left(\gamma_{1}\cup\cdots\cup\gamma_{s_{\max}}\right).$

The curves $\gamma_{s}$ have the form
$\gamma_{s}=\left\\{\left(x,\psi_{s}\left(x\right)\right):x\in\left[0,c\right]\right\\}$
for semialgebraic functions $\psi_{1},\cdots,\psi_{s_{\max}}$ of one variable.

The heart of our proof is to use the above characterization to produce
finitely many linear equations and inequalities for unknown functions
$\xi_{sk}^{l}\left(x\right)$ of one variable
($l=0,\cdots,m;k=1,\cdots,D;s=1,\cdots,s_{\max}$) with the following
properties:

(A)

If $F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{2},\mathbb{R}^{D}\right)$ is a section of
$\mathcal{H}$ then the functions

(3)
$\xi_{sk}^{l}\left(x_{1}\right)=\left.\partial_{x_{2}}^{l}F_{k}\left(x_{1},x_{2}\right)\right|_{x_{2}=\psi_{s}\left(x_{1}\right)}$

satisfy the above equations and inequalities for $x\in\left[0,c\right]$; and
conversely

(B)

If semialgebraic functions $\xi_{sk}^{l}\left(x\right)$ satisfy the above
equations and inequalities for $x\in\left[0,c\right]$, then for some small
$c^{\prime}<c$ there exists a semialgebraic section
$F=\left(F_{1},\cdots,F_{D}\right)$ of $\mathcal{H}|_{\Gamma{(c}^{\prime}{)}}$
such that (3) holds for $x\in\left[0,c^{\prime}\right]$.

We can easily deduce Theorem 1 from (A) and (B), as follows.

Because $\mathcal{H}|_{\Gamma\left(c\right)}$ has a section, (A) tells us that
the relevant equations and inequalities for the $\xi_{sk}^{l}$ admit a
solution.

Because all functions appearing in those equations and inequalities are
semialgebraic (except perhaps the unknowns $\xi_{sk}^{l}$), it follows easily
that we may take the $\xi_{sk}^{l}\left(x\right)$ to depend semialgebraically
on $x$. Thanks to (B), we obtain a semialgebraic section of
$\mathcal{H}|_{\Gamma\left(c^{\prime}\right)}$, completing the proof of
Theorem 1. See Section 7 for details.

Let us recall some of the literature regarding Problems 1, 2, 3. The
literature on Whitney’s extension problem goes back to the seminal works of H.
Whitney [41, 42], and includes fundamental contributions by G. Glaeser [31],
Yu. Brudnyi and P. Shvartsman [8, 10, 11, 9], E. Bierstone, P. Milman, and W.
Pawłucki [4, 5, 3], as well as our own papers [13, 14, 15, 16, 17, 18, 20, 19,
21, 22, 23, 24, 25, 26]. In the semialgebraic (and $o$-minimal) setting , the
analogue of the classical Whitney extension theorem is due to K. Kurdyka and
W. Pawłucki [34] and A. Thamrongthanyalak [39].

Problem 1 in the setting of $C^{1}_{loc}\left(\mathbb{R}^{n}\right)$ was
settled affirmatively by M. Aschenbrenner and A. Thamrongthanyalak [1]. Our
results on Problem 3 imply an affirmative solution for
$C^{m}_{loc}\left(\mathbb{R}^{2}\right)$. For
$C^{m}_{loc}\left(\mathbb{R}^{n}\right)$ with $m\geq 2$ and $n\geq 3$,
Problems 1, 2, 3 remain open.

The problem of deciding whether a (possibly underdetermined) system of linear
equations of the form (1) admits a $C^{0}_{loc}$ solution was proposed by
Brenner [7], and Epstein-Hochster [12]. Two independent solutions to this
problem appear in Fefferman-Kollár [27]. Fefferman-Luli [30] solved the
analogous problem for $C^{m}_{loc}$ $\left(m\geq 1\right)$. See also [29].

Kollár-Nowak [33] proved by example that an equation of the form (1) may fail
to admit a solution by $C^{0}_{loc}$-rational functions, even though
$A_{1},\cdots,A_{D}$ and $f$ are polynomials and a $C^{0}_{loc}$ solution
$\left(F_{1},\cdots,F_{D}\right)$ exists. They showed that
$x_{1}^{3}x_{2}f_{1}+(x_{1}^{3}-(1+x_{3}^{2})x_{2}^{3})f_{2}=x_{1}^{4}$ has a
continuous semialgebraic solution but no continuous rational solution
$(f_{1},f_{2})\in C^{0}_{loc}(\mathbb{R}^{3},\mathbb{R}^{2})$. However, [40]
shows that a semialgebraic $C^{0}_{loc}$ solution exists, and [33] shows that
a solution by $C^{0}_{loc}$ semialgebraic functions exists for Problems 1 and
2 posed over $\mathbb{R}^{2}$, again provided $A_{1},\cdots,A_{D},f$ are
polynomials.

A recent paper of Bierstone-Campesato-Milman [2] shows that given a system of
equations (1) with semialgebraic data $A_{i}$, $f$, there exists a function
$r:\mathbb{N}\rightarrow\mathbb{N}$ independent of $f$ such that if the system
(1) admits a $C^{r(m)}_{loc}$ solution, then it admits a semialgebraic
$C^{m}_{loc}$ solution. The result of Bierstone-Campesato-Milman is more
general than the version stated above; it applies to suitable $o$-minimal
structures.

Acknowledgement.  We are grateful to Matthias Aschenbrenner, Edward Bierstone,
Jean-Baptiste Campesato, Fushuai (Black) Jiang, Bo’az Klartag, János Kollár,
Pierre Milman, Assaf Naor, Kevin O’Neill, Wiesław Pawłucki, and Pavel
Shvartsman for their interest and valuable comments. We would also like to
thank the participants of the 11-th Whitney workshop for their interest in our
work, and we thank Trinity College Dublin, for hosting the workshop. The first
author is supported by the Air Force Office of Scientific Research (AFOSR),
under award FA9550-18-1-0069, the National Science Foundation (NSF), under
grant DMS-1700180, and the US-Israel Binational Science Foundation (BSF),
under grant 2014055. The second author is supported by NSF Grant DMS-1554733
and the UC Davis Chancellor’s Fellowship.

##  2 Notation and Preliminaries

A function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is called a Nash function
if it is real-analytic and semialgebraic.

Write $B(x,r)$ to denote the ball of radius $r$ about $x$ in $\mathbb{R}^{n}$.

The dimension of a semialgebraic set $E\subset\mathbb{R}^{n}$ is the maximum
of the dimensions of all the imbedded (not necessarily compact) submanifolds
of $\mathbb{R}^{n}$ that are contained in $E$.

We recall a few definitions from the Introduction.

Fix $m,n,D$, and a subset $E\subset\mathbb{R}^{n}$. For each $x\in E$, let

(4)
$H\left(x\right)=f\left(x\right)+I\left(x\right)\subset\mathcal{R}_{x}^{D}$

be given, where $f\left(x\right)\in\mathcal{R}_{x}^{D}$ and
$I\left(x\right)\subset\mathcal{R}_{x}^{D}$ is an $\mathcal{R}_{x}$-submodule.
Then the family

$\mathcal{H}=(H(x))_{x\in E}$

is called a bundle over $E$. $H(x)$ is called the fiber of $\mathcal{H}$ at
$x$.

When $m,n,D$ are not clear from context, we speak of a “bundle with respect to
$C^{m}_{loc}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$”.

If $\mathcal{H}$ is given by (4) and $E^{\prime}\subset E$, then we write
$\left.\mathcal{H}\right|_{E^{\prime}}$ to denote the bundle
$\left(H\left(x\right)\right)_{x\in E^{\prime}}$, and refer to it as the
restriction of $\mathcal{H}$ to $E^{\prime}$. If $\mathcal{H}=(H(x))_{x\in E}$
and $\mathcal{H}^{\prime}=(H^{\prime}(x))_{x\in E}$ are bundles,
$\mathcal{H}^{\prime}$ is called a subbundle of $\mathcal{H}$ if
$H^{\prime}(x)\subset H(x)$ for all $x\in E$. We write
$\mathcal{H}\supset\mathcal{H}^{\prime}$ to denote that $\mathcal{H}^{\prime}$
is a subbundle of $\mathcal{H}$.

What we called a “bundle” in [28] we now call a “classical bundle”.

The definition is as follows. Fix $m,n,D$. Let $E\subset\mathbb{R}^{n}$ be
compact. A classical bundle over $E$ is a family
$\mathcal{{H}}=\left({H}\left(x\right)\right)_{x\in E}$ of (possibly empty)
affine subspaces ${H}\left(x\right)\subset\mathcal{P}^{D}$, parametrized by
the points $x\in E$, such that each non-empty ${H}\left(x\right)$ has the form

${H}\left(x\right)=\vec{P}^{x}+\vec{I}\left(x\right)$

for some $\vec{P}^{x}\in\mathcal{P}^{D}$ and some $\mathcal{R}_{x}$-submodule
$\vec{I}\left(x\right)$ of $\mathcal{P}^{D}$.

When $m,n,D$ are not clear from context, we speak of a “classical bundle with
respect to $C^{m}(\mathbb{R}^{n},\mathbb{R}^{D})$”.

We remark again that our notion of bundle differs from the notion of bundles
considered previously (e.g., [28]). In the present version, we do not require
that $E$ be compact and we require all the fibers $H(x)$ to be non-empty.

A section of the bundle $\mathcal{H}$ is a vector-valued function $F\in
C_{loc}^{m}(\mathbb{R}^{n},\mathbb{R}^{D})$ such that $J_{x}F\in H(x)$ for all
$x\in E$. A section of a classical bundle $\mathcal{H}$ is a vector-valued
function $F\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{D})$ such that $J_{x}F\in
H(x)$ for all $x\in E$.

##  3 Tools

###  3.1 Glaeser Refinements, Stable Glaeser Refinements

Given a bundle $\mathcal{H}=(H(x))_{x\in E}$ for
$C^{m}_{loc}(\mathbb{R}^{n},\mathbb{R}^{D})$ or a classical bundle
$\mathcal{H}=(H(x))_{x\in E}$ for $C^{m}(\mathbb{R}^{n},\mathbb{R}^{D})$, we
define the Glaeser refinement $\mathcal{H}^{\prime}=(H^{\prime}(x))_{x\in E}$
as follows:

(GR)

Let $x_{0}\in E$. A given $P_{0}\in H(x_{0})$ belongs to $H^{\prime}(x_{0})$
if and only if the following holds. Given $\epsilon>0$, there exists
$\delta>0$ such that for all $x_{1},\cdots,x_{k}\in B(x_{0},\delta)\cap E$,
where $k$ is a large enough constant depending only on $m$, $n$, and $D$,
there exist $P_{i}\in H(x_{i})$ ($i=1,\cdots,k$), such that

$\left|\partial^{\alpha}(P_{i}-P_{j})(x_{i})\right|\leq\epsilon|x_{i}-x_{j}|^{m-|\alpha|},$

for all $|\alpha|\leq m,0\leq i,j\leq k$.

A bundle or a classical bundle $\mathcal{H}$ is Glaeser stable if
$\mathcal{H}^{\prime}=\mathcal{H}$.

Note that the Glaeser refinement $\mathcal{H}^{\prime}$ of $\mathcal{H}$ may
have empty fibers, even if $\mathcal{H}$ has none. In that case, we know that
$\mathcal{H}$ has no sections. If $\mathcal{H}$ is a classical bundle, then so
is $\mathcal{H}^{\prime}$. If $\mathcal{H}$ is a bundle and no fibers of
$\mathcal{H}^{\prime}$ are empty, then $\mathcal{H}^{\prime}$ is a bundle.
Both for bundles and for classical bundles, every section of $\mathcal{H}$ is
a section of $\mathcal{H}^{\prime}$. (See [28] for the case of classical
bundles; the elementary proofs carry over unchanged for bundles.) Note in
particular that if a given bundle $\mathcal{H}$ has a section, then
$\mathcal{H}^{\prime}$ has no empty fibers, hence $\mathcal{H}^{\prime}$ is a
bundle and $\mathcal{H}^{\prime}$ has a section.

Starting from a classical bundle $\mathcal{H}$, or a bundle $\mathcal{H}$ with
a section, we can perform iterated Glaeser refinement to pass to ever smaller
subbundles $\mathcal{H}^{\left(1\right)}$, $\mathcal{H}^{\left(2\right)}$,
etc., without losing sections. We set
$\mathcal{H}^{\left(0\right)}=\mathcal{H}$, and for $l\geq 0$, we set
$\mathcal{H}^{\left(l+1\right)}=\left(\mathcal{H}^{\left(l\right)}\right)^{\prime}$.
Thus, by an obvious induction on $l$, we have
$\mathcal{H=\mathcal{H}}^{\left(0\right)}\supset\mathcal{H}^{\left(1\right)}\supset\cdots$,
yet $\mathcal{H}$ and $\mathcal{H}^{\left(l\right)}$ have the same sections
for all $l\geq 0$.

If $\mathcal{H}=(H(x))_{x\in E}$ is a semialgebraic bundle with respect to
$C^{m}_{loc}(\mathbb{R}^{n},\mathbb{R}^{D})$, by an obvious induction on $l$,
we have $H^{(l)}(x)$ depends semialgebraically on $x$, where
$\mathcal{H}^{(l)}=(H^{(l)}(x))_{x\in E}.$

In principle, each $\mathcal{H}^{\left(l\right)}$ can be computed from
$\mathcal{H}$. We remark that iterated Glaeser refinement stabilizes after
finitely many iterations (i.e. for a large enough integer $l^{*}$ determined
by $m,n,D$, we have $\mathcal{H}^{(l^{*}+1)}=\mathcal{H}^{(l^{*})}$; thus
$\mathcal{H}^{(l^{*})}$ is Glaeser stable. See [28] for the case of classical
bundles; the argument, which goes back to Glaeser [31] and Bierstone-Milman-
Pawłucki [4, 5], applies unchanged for bundles. We call
$\mathcal{H}^{(l^{*})}$ the stable Glaeser refinement of $\mathcal{H}$.)

The main results of [28] give the following

###### Theorem 2

For a large enough integer constant $l_{\ast}$ determined by $m,n,$ and $D$,
the following holds. Let $\mathcal{H}$ be a classical bundle with respect to
$C^{m}\left(\mathbb{R}^{n},\mathbb{R}^{D}\right)$. Let
$\mathcal{H}^{\left(0\right)},\mathcal{H}^{\left(1\right)},\mathcal{H}^{\left(2\right)},\cdots$
be its iterated Glaeser refinements. Then $\mathcal{H}$ has a section if and
only if $\mathcal{H}^{\left(l_{\ast}\right)}$ has no empty fibers. Suppose
$\mathcal{H}^{\left(l_{\ast}\right)}$ has no empty fibers. Let $x_{0}\in E$
and let $P_{0}$ belong to the fiber of $\mathcal{H}^{\left(l_{\ast}\right)}$
at $x_{0}$. Then there exists a section $F$ of the bundle $\mathcal{H}$, such
that $J_{x_{0}}(F)=P_{0}$. Moreover, there exists a constant $k^{\\#}$
depending only on $m,n,$ and $D$ such that the following holds: Suppose
$\mathcal{H}=(H(x))_{x\in E}$ is a Glaeser stable classical bundle. Assume the
following holds for some constant $M>0$:

  * •

Given $x_{1},\cdots x_{k^{\\#}}\in E$, there exist polynomials
$P_{1},\cdots,P_{k^{\\#}}\in\mathcal{P}^{D}$, with $P_{i}\in H(x_{i})$ for
$1\leq i\leq k^{\\#}$; $|\partial^{\alpha}P_{i}(x_{i})|\leq M$ for all
$|\alpha|\leq m,1\leq i\leq k^{\\#}$; and
$|\partial^{\alpha}(P_{i}-P_{j})(x_{j})|\leq M|x_{i}-x_{j}|^{m-|\alpha|}$ for
all $|\alpha|\leq m,1\leq i,j\leq k^{\\#}$.

Then there exists $F\in C^{m}(\mathbb{R}^{n},\mathbb{R}^{D})$ with
$\|F\|_{C^{m}(\mathbb{R}^{n},\mathbb{R}^{D})}\leq C(m,n,D)M$ and $J_{x}(F)\in
H(x)$ for all $x\in E$.

###  3.2 Puiseux Series

We will use the following elementary result regarding semialgebraic functions.
For a proof, see [32].

###### Lemma 3.1

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is semialgebraic. Then there
exists a polynomial $P\left(z,x\right)\not\equiv 0$ on $\mathbb{R}^{2}$ such
that $P\left(f\left(x\right),x\right)\equiv 0$. Moreover, for each
$x_{0}\in\mathbb{R}$ there exists $\delta>0$ such that $f\left(x\right)$ for
$x\in(x_{0},x_{0}+\delta)$ is given by a convergent Puiseux series.

###### Corollary 3.1

Let $F(x)$ be a semialgebraic function of one variable, satisfying
$|F(x)|=O(x^{p})$ on $(0,c]$ for some given $p$. Then the derivatives of $F$
satisfy $|F^{(k)}(x)|=O(x^{p-k})$ on $(0,c^{\prime}]$ for some $c^{\prime}$.
Similarly, if $F(x)=o(x^{p})$ for $x$ in $(0,c)$, then $F^{(k)}(x)=o(x^{p-k})$
for $x$ in $(0,c^{\prime})$. More generally, $|F^{(k)}(x)|=O(|F(x)|/x^{k})$ on
$(0,c^{\prime})$.

###### Corollary 3.2

Let $F$ be a semialgebraic function in $C^{m}_{loc}(\Omega_{1})$, where
$\Omega_{\delta}=\\{(x,y)\in\mathbb{R}^{2}:0\leq y\leq x<\delta\\}$ for
$\delta>0$. Then for small enough $\delta$, $F|_{\Omega_{\delta}}$ extends to
a $C^{m}$ semialgebraic function on $\mathbb{R}^{2}$.

Sketch of proof. The result follows in one line from known results, but we
sketch an elementary proof.

Without loss of generality, we may suppose that $J_{(0,0)}F=0$. Then
$\partial_{x_{2}}^{k}F(x_{1},0)=o(x_{1}^{m-k})$ for $k\leq m$, hence
$\partial_{x_{1}}^{l}\partial_{x_{2}}^{k}F(x_{1},0)=o(x_{1}^{m-k-l})$ for
$0\leq k,l\leq m$.

We set $\tilde{F}(x_{1},x_{2})$ equal to the m-th degree Taylor polynomial of
$x_{2}\mapsto F(x_{1},x_{2})$ about $x_{2}=0$ for each fixed $x_{1}$. The
above estimates for derivatives of $F$ show that $\tilde{F}$ is $C^{m}$ on
$\tilde{\Omega}_{\delta}=\\{(x_{1},x_{2}):0\leq-x_{2}\leq x_{1}\leq\delta\\}$,
and its $x_{2}$-derivatives up to order $m$ agree with those of $F$ on the
$x_{1}$-axis. In particular, $J_{(0,0)}\tilde{F}=0$.

Similarly, we set $F^{\\#}(x_{1},x_{2})$ equal to the m-th degree Taylor
polynomial of $x_{2}\mapsto F(x_{1},x_{2})$ about $x_{2}=x_{1}$ for each fixed
$x_{1}$. Then $F^{\\#}$ is $C^{m}$ on
$\Omega^{\\#}_{\delta}=\\{(x_{1},x_{2}):0\leq x_{1}\leq x_{2}\leq 2x_{1}\leq
2\delta\\}$, and its $x_{2}$-derivatives up to order $m$ agree with those of
$F$ on the line $x_{1}=x_{2}$. In particular, $J_{(0,0)}F^{\\#}=0$.

Setting $F^{+}=\begin{cases}F&\mbox{on }\Omega_{\delta}\\\ \tilde{F}&\mbox{on
}\tilde{\Omega}_{\delta}\\\ F^{\\#}&\mbox{on
}\Omega^{\\#}_{\delta}\end{cases}$, we see that $F^{+}$ is a $C^{m}$
semialgebraic function on $\\{(x_{1},x_{2}):x_{1}\in[0,\delta],-x_{1}\leq
x_{2}\leq 2x_{1}\\},F^{+}=F$ on $\Omega_{\delta}$, and $J_{(0,0)}F^{+}=0$.

Next, let $\theta(t)$ be a $C^{m}$ semialgebraic function of one variable,
equal to 1 in $[0,1]$ and supported in $[-1,2]$. Then, for small enough
$\delta$, the function $F^{++}(x_{1},x_{2})=\theta(\frac{x_{2}}{x_{1}})\cdot
F^{+}(x_{1},x_{2})$ for $x_{1}>0$, $F^{++}(x_{1},x_{2})=0$ otherwise, is a
$C^{m}$ semialgebraic function on the disc $B(0,\delta)$ that agrees with our
given $F$ on $\Omega_{\delta}$.

Finally, multiplying $F^{++}$ by a semialgebraic cutoff function supported in
a small disc about $(0,0)$ and equal to $1$ in a smaller disc, we obtain a
$C^{m}$ semialgebraic function on $\mathbb{R}^{2}$ that agrees with $F$ on
$\Omega_{\delta}$ for small enough $\delta$.

###  3.3 Singularities of Semialgebraic Sets and Functions

We recall a few standard properties of semialgebraic sets and functions.

  * •

Let $U\subset\mathbb{R}^{n}$ be an open semialgebraic set, and let
$F:U\rightarrow\mathbb{R}^{k}$ be semialgebraic. Then there exists a
semialgebraic subset $X\subset U$ of dimension less than $n$ (the “singular
set” of $F$) such that $F$ is real-analytic on $U\setminus X$. (See Chapter 8
in [6].)

  * •

A zero-dimensional semialgebraic set is finite. A one-dimensional
semialgebraic set is a union of finitely many real-analytic arcs and finitely
many points. (See Chapter 2 in [6].)

###  3.4 Existence of Semialgebraic Selections

For sets $X,Y$, we denote a map $\Xi$ from $X$ to the power set of $Y$ by
$\Xi:X\rightrightarrows Y$ and call such $\Xi$ a set-valued map; a set-valued
map $\Xi$ is semialgebraic if $\\{(x,y):y\in\Xi(x)\\}$ is a semialgebraic set.
Let $E\subset\mathbb{R}^{n}$ and $\Xi:E\rightrightarrows\mathbb{R}^{D}$. A
selection of $\Xi$ is a map $f:E\rightarrow\mathbb{R}^{D}$ such that
$f(x)\in\Xi(x)$ for every $x\in E$. We recall the following well-known result
regarding semialgebraic selection (see, for example, [36]).

###### Theorem 3

Let $\Xi:E\rightrightarrows\mathbb{R}^{D}$ be semialgebraic. If each $\Xi(x)$
is nonempty, then $\Xi$ has a semialgebraic selection.

###  3.5 Growth of Semialgebraic Functions

Recall from [30] the following result

######  Lemma 3.2 (Growth Lemma)

Let $E\subset\mathbb{R}^{n_{1}}$ and $E^{+}\subset E\times\mathbb{R}^{n_{2}}$
be compact and semialgebraic, with $\dim E^{+}\geq 1$. Let $A$ be a
semialgebraic function on $E^{+}$. Then there exist an integer $K\geq 1$, a
semialgebraic function $A_{1}$ on $E$, and a compact semialgebraic set
$\underline{E}^{+}\subset E^{+}$, with the following properties.

(GL1)

$\dim\underline{E}^{+}<\dim E^{+}$.

For $x\in E$, set
$E^{+}\left(x\right)=\left\\{y\in\mathbb{R}^{n_{2}}:\left(x,y\right)\in
E^{+}\right\\}$ and
$\underline{E}^{+}\left(x\right)=\left\\{y\in\mathbb{R}^{n_{2}}:\left(x,y\right)\in\underline{E}^{+}\right\\}$.
Then, for each $x\in E$, the following hold.

(GL2)

If $\underline{E}^{+}\left(x\right)$ is empty, then

$\left|A\left(x,y\right)\right|\leq A_{1}\left(x\right)\text{ for all }y\in
E^{+}\left(x\right).$

(GL3)

If $\underline{E}^{+}\left(x\right)$ is non-empty, then

$\left|A\left(x,y\right)\right|\leq
A_{1}\left(x\right)\cdot\left[\text{dist}\left(y,\underline{E}^{+}\left(x\right)\right)\right]^{-K}\text{
for all }y\in E^{+}\left(x\right)\setminus\underline{E}^{+}\left(x\right).$

The Growth Lemma follows easily from a special of a theorem of Łojasiewicz and
Wachta [35], as explained in [30]. We thank W. Pawłucki for teaching us that
implication.

We will apply the Growth Lemma to prove the following.

###### Lemma 3.3

Let $F\left(x,y\right)$ be a bounded semialgebraic function on
$\left[-1,1\right]\times(0,1],$ and suppose that

(5) $\lim_{y\rightarrow 0^{+}}F\left(x,y\right)=0\text{ for each
}x\in\left[-1,1\right]\text{.}$

Then there exist a positive integer $N$ and a semialgebraic function
$A\left(x\right)$ on $\left[-1,1\right]$ such that

$F\left(x,y\right)\leq A\left(x\right)y^{\frac{1}{N}}\text{ for all
}\left(x,y\right)\in\left[-1,1\right]\times(0,1]\text{.}$

Proof. It is enough to show that for some positive integer $N$ we have

(6)
$\sup_{y\in(0,1]}\frac{\left|F\left(x,y\right)\right|}{y^{1/N}}<\infty\text{
for all }x\in\left[-1,1\right]\text{,}$

for we may then set
$A\left(x\right)=\sup_{y\in(0,1]}\frac{\left|F\left(x,y\right)\right|}{y^{1/N}}$,
and $A\left(x\right)$ will depend semialgebraically on $x$.

For each fixed $x$, the function $y\mapsto F\left(x,y\right)$ is bounded and
given near $\left(0,0\right)$ by a convergent Puiseux series that tends to
zero as $y\rightarrow 0^{+}$. Hence, for some positive integer $N_{x}$ we have

(7)
$\sup_{y\in(0,1]}\frac{\left|F\left(x,y\right)\right|}{y^{1/N_{x}}}<\infty\text{.}$

Our task is to show that $N_{x}$ may be taken independent of $x.$ Thanks to
(7), we may exclude from consideration any given finite set of “bad”
$x\in\left[-1,1\right]$.

We recall our main hypothesis (5). For each
$\left(x,\varepsilon\right)\in\left[-1,1\right]\times(0,1]$ there exists
$\delta\in(0,1]$ such that $\left(x,\varepsilon,\delta\right)$ belongs to the
semialgebraic set

$\left\\{\left(x,\varepsilon,\delta\right)\in\left[-1,1\right]\times(0,1]\times(0,1]:\left|F\left(x,y\right)\right|\leq\varepsilon\text{
for all }y\in(0,\delta]\right\\}.$

Hence, there exists a semialgebraic function
$\delta\left(x,\varepsilon\right)$ mapping $\left[-1,1\right]\times(0,1]$ into
$(0,1]$ such that

(8) $\left|F\left(x,y\right)\right|\leq\varepsilon\text{ for
}y\in(0,\delta\left(x,\varepsilon\right)],x\in\left[-1,1\right],\varepsilon\in(0,1].$

We set $\delta\left(x,0\right)=1$ for $x\in\left[-1,1\right]$. Then
$\delta:\left[-1,1\right]\times\left[0,1\right]\rightarrow(0,1]$ is
semialgebraic and satisfies (8).

We now apply Lemma 3.2 to the function
$\frac{1}{\delta\left(x,\varepsilon\right)}$.

Thus, we obtain a semialgebraic set
$\underline{E}\subset\left[-1,1\right]\times\left[0,1\right]$, a positive
integer $N,$ and a positive semialgebraic function
$\underline{\delta}\left(x\right)$ on $\left[-1,1\right]$, with the following
properties.

  * •

$\dim\underline{E}\leq 1$.

  * •

For $x\in\left[-1,1\right]$, let
$\underline{E}\left(x\right)=\left\\{\varepsilon:\left(x,\varepsilon\right)\in\underline{E}\right\\}$.

Then

(9)
$\delta\left(x,\varepsilon\right)\geq\underline{\delta}\left(x\right)\text{
(all }\varepsilon>0\text{) if }\underline{E}=\emptyset$

and

(10)
$\delta\left(x,\varepsilon\right)\geq\underline{\delta}\left(x\right)\cdot\left[\text{dist}\left(\varepsilon,\underline{E}\left(x\right)\right)\right]^{N}\text{
(all }\varepsilon\not\in\underline{E}(x)\text{) if
}\underline{E}\not=\emptyset\text{.}$

Because $\dim\underline{E}\leq 1,$ there are at most finitely many
$x\in\left[-1,1\right]$ for which $\underline{E}\left(x\right)$ is infinite.

As explained above, we may discard those “bad” $x$, it is enough to prove (6)
for all $x$ such that $\underline{E}\left(x\right)$ is finite.

From now on, we restrict attention to “good” $x,$ i.e., those $x$ for which
$\underline{E}\left(x\right)$ is finite.

Set

$\underline{\mathcal{\varepsilon}}\left(x\right)=\left\\{\begin{array}[]{l}\frac{1}{2}\min\left(\underline{E}\left(x\right)\setminus\left\\{0\right\\}\right)\\\
1\end{array}\right.\begin{array}[]{l}\text{if
}\underline{E}\left(x\right)\text{ contains points other than }0\\\
\text{otherwise}\end{array}\text{.}$

So $\underline{\mathcal{\varepsilon}}\left(x\right)>0$ for all “good” $x$.

If $\underline{E}\left(x\right)\not=\emptyset$, then
$\text{dist}\left(\varepsilon,\underline{E}\left(x\right)\right)\geq\varepsilon$
for $0<\varepsilon\leq\underline{\mathcal{\varepsilon}}\left(x\right)$, hence
(10) gives

(11)
$\delta\left(x,\varepsilon\right)\geq\underline{\delta}\left(x\right)\varepsilon^{N}\text{
for }0<\varepsilon\leq\underline{\varepsilon}\left(x\right)\text{.}$

If instead $\underline{E}\left(x\right)=\emptyset$, then because
$\underline{\mathcal{\varepsilon}}\left(x\right)=1,$ (9) again gives (11).
Thus, (11) holds in all cases.

Now suppose
$0<y<\underline{\delta}\left(x\right)\cdot\left(\underline{\varepsilon}\left(x\right)\right)^{N}$.

Then, setting
$\varepsilon=\left(\frac{y}{\underline{\delta}\left(x\right)}\right)^{1/N}$
and applying (11), we find that $\delta\left(x,\varepsilon\right)\geq y.$ The
defining property of $\delta\left(x,\varepsilon\right)$ therefore tells us
that

$\left|F\left(x,y\right)\right|\leq\varepsilon=\left(\frac{y}{\underline{\delta}\left(x\right)}\right)^{1/N}\text{.}$

Thus, for any “good” $x,$ we have shown that

(12)
$\frac{\left|F\left(x,y\right)\right|}{y^{1/N}}\leq\left(\underline{\delta}\left(x\right)\right)^{-1/N}\text{
for
}0<y<\underline{\delta}\left(x\right)\cdot\left(\underline{\varepsilon}\left(x\right)\right)^{N}\text{.}$

On the other hand, recall that $F$ is bounded; say,
$\left|F\left(x,y\right)\right|\leq M$ for all
$\left(x,y\right)\in\left[-1,1\right]\times(0,1]$.

Hence,

(13)
$\frac{\left|F\left(x,y\right)\right|}{y^{1/N}}\leq\frac{M}{\left(\underline{\delta}\left(x\right)\right)^{1/N}\underline{\varepsilon}\left(x\right)}\text{
for
}\underline{\delta}\left(x\right)\cdot\left(\underline{\varepsilon}\left(x\right)\right)^{N}\leq
y\leq 1\text{.}$

Our desired estimate (6) is now immediate from (12) and (13).

The proof of Lemma 3.3 is complete.

Similar ideas can be used to prove an $n$-dimensional version of Lemma 3.3,
but we don’t discuss it here.

###  3.6 Logarithmic Derivatives of Semialgebraic Functions

Let $V$ be a semialgebraic subset of $\mathbb{R}^{n}\times\mathbb{R}^{m}$.
Given $x\in\mathbb{R}^{n}$, we write $V(x)$ to denote the set of all
$t\in\mathbb{R}^{m}$ such that $(x,t)\in V$. Given
$(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$, we write $\delta_{V}(x,t)$ to
denote the distance from $t$ to $V(x)$. We take $\delta_{V}(x,t)=+\infty$ if
$V(x)$ is empty. For a smooth function $F(x,t)$ on
$\mathbb{R}^{n}\times\mathbb{R}^{m}$, we write $\nabla_{t}F(x,t)$ to denote
the gradient of the function $t\mapsto F(x,t)$.

The following theorem is proven by A. Parusinski in [37, 38]. We thank Edward
Bierstone, Jean-Baptiste Campesato, Pierre Milman, and Wieslaw Pawłucki for
pointing out the references, and thus helping us remove 10 pages from our
paper.

###### Theorem 4

Let $F(x,t)$ be a (real-valued) subanalytic function of
$(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$. Then there exist a closed
codimension 1 subanalytic set $V\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$ and
a constant $C>0$ such that outside $V$ the function $F$ is smooth and
moreover,

(14) $|\nabla_{t}F(x,t)|\leq
C\frac{\left|F\left(x,t\right)\right|}{\delta_{V}\left(x,t\right)}\text{.}$

If $F$ is semialgebraic, then we can take $V$ to be semialgebraic.

As a special case of Theorem 4, we have the following.

###### Theorem 5

Let $F\left(x\right)$ be a semialgebraic function on $\mathbb{R}^{n}$. Then
there exist a closed semialgebraic $V\subset\mathbb{R}^{n}$ of dimension at
most $\left(n-1\right)$, and a constant $C$, such that $F$ is $C^{m}_{loc}$
outside $V$, and

$\left|\nabla F\left(x\right)\right|\leq
C\left|F\left(x\right)\right|\cdot\left[\text{dist}\left(x,V\right)\right]^{-1}$

for $x\in\mathbb{R}^{n}\setminus V$.

###  3.7 Variant of Helly’s Theorem

We recall the following result from convex geometry. Surely more precise
versions of the result are well known, but we had trouble tracking down a
reference so we will provide a proof.

######  Theorem 6 (Helly’s Theorem Variant)

Let $(p_{\omega})_{\omega\in\Omega}$ be a family of seminorms on a vector
space $V$ of dimension $D$. Assume that
$\sup_{\omega\in\Omega}p_{\omega}(v)<\infty$ for every $v\in V$. Then there
exist $\omega_{1},\cdots,\omega_{L}\in\Omega$, with $L$ depending only on $D$,
such that

$\sup_{\omega\in\Omega}p_{\omega}(v)\leq
C\cdot\max\\{p_{\omega_{1}}(v),\cdots,p_{\omega_{L}}(v)\\}\text{ for all }v\in
V,$

with $C$ also depending only on $D$.

We use the following variant of the classical Helly theorem (see Section 3 in
[14]) from elementary convex geometry.

###### Lemma 3.4

Let $(K_{\omega})_{\omega\in\Omega}$ be a collection of compact convex
symmetric subsets of $\mathbb{R}^{D}$. Suppose the intersection of all the
$K_{\omega}$ has nonempty interior. Then there exist
$\omega_{1},\cdots,\omega_{L}$ such that $K_{\omega_{1}}\cap\cdots\cap
K_{\omega_{L}}\subset C\cdot\bigcap_{\omega\in\Omega}K_{\omega}$, where $C$
and $L$ depend only on $D$.

The proof of the “Lemma on Convex Sets” in Section 3 of [14] applies here and
proves Lemma 3.4, even though our present hypotheses differ slightly from
those of [14].

We apply Lemma 3.4 to prove Theorem 6.

Proof of Theorem 6. Suppose first that each $p_{\omega}$ is a norm, not just a
seminorm. Then the conclusion of Theorem 6 follows by applying Lemma 3.4 to
the family of convex sets $K_{\omega}=\\{v\in V:p_{\omega}(v)\leq 1\\}$,
${\omega\in\Omega}$.

Now suppose each $p_{\omega}$ is a seminorm. Let $H(\omega)=\\{v\in
V:p_{\omega}(v)=0\\}$, and let $H$ be the intersection of all the $H(\omega)$.
Each $H(\omega)$ is a vector subspace of $V$. Consequently there exist
$\lambda_{1},\cdots,\lambda_{s}\in\Omega$, with $s\leq D$, such that
$H=H(\lambda_{1})\cap\cdots\cap H(\lambda_{s})$.

For $\omega\in\Omega$ and $v\in V$, set
$p^{*}_{\omega}(v)=p_{\lambda_{1}}(v)+\cdots+p_{\lambda_{s}}(v)+p_{\omega}(v)$.
Then $p^{*}_{\omega}$ is a seminorm on $V$, and $p^{*}_{\omega}(v)=0$ if and
only if $v\in H$. Regarding each $p^{*}_{\omega}$ as a norm on $V/H$, and
applying Theorem 6 for collections of norms, we complete the proof of Theorem
6.

##  4 Preliminary Reductions

The purpose of this section is to reduce Theorem 1 to the following:

######  Lemma 4.1 (Main Lemma)

Let $\mathcal{H}=(H(x))_{x\in\mathbb{R}^{2}}$ be a semialgebraic bundle for
$C^{m}_{loc}(\mathbb{R}^{2},\mathbb{R}^{D})$. Assume $\mathcal{H}$ is Glaeser
stable. Assume $H(0)=\\{0\\}$. Then, for small enough $c>0$,
$\mathcal{H}|_{\Gamma(c)}$ has a semialgebraic section, where
$\Gamma(c)=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}\in\left[0,c\right],0\leq
x_{2}\leq x_{1}\\}.$

To deduce Theorem 1 from Lemma 4.1 we argue as follows.

Suppose we are given a Glaeser stable bundle
$\mathcal{H}=(H(x))_{x\in\mathbb{R}^{2}}$ for
$C^{m}_{loc}(\mathbb{R}^{2},\mathbb{R}^{D})$ with $H(x)\subset\mathcal{P}^{D}$
depending semialgebraically on $x$. Assume $H(0)=\\{0\\}$.

Let
$\Gamma(c)=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{1}\in\left[0,c\right],0\leq
x_{2}\leq x_{1}\\}$. Theorem 2 tells us that $\mathcal{H}|_{\Gamma(c)}$ has a
section $F_{c}$. The main lemma asserts that for $c$ small enough
$\mathcal{H}|_{\Gamma(c)}$ has a semialgebraic section.

We will cover a full neighborhood of $0$ by rotating wedges of the form
$\Gamma(c)$. Using a partition of unity subordinate to the cover and the fact
that $H(0)=\\{0\\}$, we can then patch together sections of $\mathcal{H}$, and
obtain a semialgebraic section over a full neighborhood of $0$.

We may drop the restriction $H(0)=\\{0\\}$, because without loss of generality
our given section $F_{c}$ has jet $0$ at the origin, so we may just cut down
$H(0)$ to $\\{0\\}$. We can also drop the restriction that $\mathcal{H}$ is
Glaeser stable (assuming $\mathcal{H}$ has a section) since we can always pass
to the stable Glaeser refinement. Thus, any semialgebraic bundle having a
section has a semialgebraic section over some neighborhood of $0$. We can use
compactness and a partition of unity to conclude that $\mathcal{H}$ admits a
semialgebraic section over any given compact set.

###### Lemma 4.2

Suppose $H(z)$ depends semialgebraically on $z\in\mathbb{R}^{2}$. If
$\mathcal{H}=(H(z))_{z\in\mathbb{R}^{2}}$ has a section, then $\mathcal{H}$
has a section $F\in C^{m}_{loc}(\mathbb{R}^{2},\mathbb{R}^{D})$ such that for
all $|\alpha|\leq m$, $|\partial^{\alpha}F(x)|\leq C(1+|x|)^{K}$ on
$\mathbb{R}^{2}$, for some $C$ and $K$.

Proof. To prove this lemma, we may assume that $\mathcal{H}$ is Glaeser
stable.

Taking $E_{R}=\left\\{x\in\mathbb{R}^{2}:\left|x\right|\leq R\right\\}$ with
$R\geq 1$, and applying Theorem 2, we obtain a section $F_{R}$ of
$\mathcal{H}|_{E_{R}}$, with $\left|\left|F_{R}\right|\right|_{C^{m}}\leq
C\left(R\right)^{K}$, because the “$M$ ” in the result quoted above applied to
$\mathcal{H}|_{E_{R}}$ can be taken to depend semialgebraically on $R$.
(That’s where we use the fact that the bundle $\mathcal{H}$ is semialgebraic.)

We can now easily use a partition of unity to patch together $F_{2^{k}}$,
$k=1,2,3,\cdots$, into a section $F$ as in the conclusion of Lemma 4.2.

Fix $K$ as in the conclusion of Lemma 4.2. Let $\Phi:\text{ Open Disc
}\Delta\rightarrow\mathbb{R}^{2}$ be a semialgebraic diffeomorphism, for
example, $\Phi(x)=\frac{x}{1-|x|^{2}}$. Let $\theta(x)>0$ be a semialgebraic
function on $\mathbb{R}^{2}$ that tends to zero so rapidly that

$\partial^{\alpha}[(\theta F)\circ\Phi](y)\rightarrow 0\text{, for all
}|\alpha|\leq m\text{ as }y\rightarrow\partial\Delta,$

whenever $|\partial^{\alpha}F(x)|\leq C(1+|x|)^{K}$ on $\mathbb{R}^{2}$,
$|\alpha|\leq m$.

We can now form a bundle $\mathcal{H}^{*}$ as follows: For $x$ in $\Delta$,
the fiber $H^{*}(x)$ consists of all $J_{x}((\theta F)\circ\Phi)$ for sections
$F$ of the bundle $\mathcal{H}$.

The fibers of $\mathcal{H}^{*}$ over points not in $\Delta$ are $\\{0\\}$.

Then $\mathcal{H}^{*}$ is a semialgebraic bundle admitting a section.

We have seen that semialgebraic bundles with sections have semialgebraic
sections over any compact set. In particular, $\mathcal{H}^{*}$ has a
semialgebraic section $\mathcal{F}$ over $\Delta^{\text{closure}}$. Then
$\frac{\mathcal{F}\circ\Phi^{-1}(x)}{\theta(x)}$ is a semialgebraic section of
$\mathcal{H}$ over $\mathbb{R}^{2}$.

Consequently, we can deduce Theorem 1 from Lemma 4.1.

The rest of the paper is devoted to the proof of Lemma 4.1.

##  5 Characterization of Sections

###  5.1 Semialgebraic Bundles

Fix $U\subset\mathbb{R}^{n}$ open, semialgebraic. Fix
$\psi:U\rightarrow\mathbb{R}^{k}$ Nash. Let
$\hat{\psi}(x)=(x,\psi(x))\in\mathbb{R}^{n}\times\mathbb{R}^{k}$ for $x\in U$.
We set $\hat{U}=\hat{\psi}(U)$. Let $\mathcal{P}$ denote the vector space of
polynomials of degree at most $m$ on $\mathbb{R}^{n}\times\mathbb{R}^{k}$. We
write $z=(x,y)$ to denote a point of $\mathbb{R}^{n}\times\mathbb{R}^{k}$. We
write $\mathcal{R}_{z}$ to denote the ring obtained from $\mathcal{P}$ by
multiplication of $m$-jets at $z$. We fix a bundle
$\mathcal{H}=(H(z))_{z\in\hat{U}}$, where, for each
$z=\hat{\psi}(x)\in\hat{U}$ we have $H(z)=f^{x}+I(x)$,
$f^{x}\in\mathcal{P}^{D}$, $I(x)$ an $\mathcal{R}_{\hat{\psi}(x)}$-submodule
of $\mathcal{P}^{D}$. (We point out that $\mathcal{H}$ is a bundle, not a
classical bundle, see Remark 1.1.)

We suppose $\mathcal{H}$ is Glaeser stable. We assume that $H(z)$ depends
semialgebraically on $z\in\hat{U}$. (We sometimes abuse notion by writing
$I(z)$ for $I(x)$, where $z=\hat{\psi}(x)$.)

Under the above assumptions and definitions, we will prove the following
result.

###### Lemma 5.1

There exist a semialgebraic set $U_{\text{bad}}\subset\mathbb{R}^{n}$ of
dimension less than $n$; Nash functions $A_{j\beta}^{i},G^{i}$ on $U\setminus
U_{\text{bad}}$ ($i=1,\cdots,i_{\max},j=1,\cdots,D,\beta$ a multiindex of
order $\leq m$ for $\mathbb{R}^{k}$) with the following property. Let
$B\subset U\setminus U_{\text{bad}}$ be a closed ball. Set
$\hat{B}=\hat{\psi}(B)$. Let $F=(F_{1},\cdots,F_{D})\in
C^{m}_{loc}(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D})$. Then $F$ is a
section of $\mathcal{H}|_{\hat{B}}$ if and only if $\sum_{|\beta|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}(x)\cdot(\partial_{y}^{\beta}F_{j}(x,\psi(x)))=G^{i}(x)$
for all $x\in B$ (each $i$).

Proof. We may suppose that $f^{x}$ and $I(x)$ depend semialgebraically on
$x\in U$. We write $f^{x}=(f_{1}^{x},\cdots,f_{D}^{x})$ and
$\psi(x)=(\psi_{1}(x),\cdots,\psi_{k}(x))\quad(x\in U)$.

For $l=1,\cdots,n$, we introduce the vector field

$X_{l}=\frac{\partial}{\partial
x_{l}}+\sum_{p=1}^{k}\frac{\partial\psi_{p}(x)}{\partial
x_{l}}\frac{\partial}{\partial y_{p}}\text{on }U\times\mathbb{R}^{k}.$

On $U\times\mathbb{R}^{k}$, then $X_{l}$ are Nash, and
$[X_{l},X_{l^{\prime}}]=0$. For $\alpha=(\alpha_{1},\cdots,\alpha_{n})$, we
write $X^{\alpha}=X_{1}^{\alpha_{1}}\cdots X_{n}^{\alpha_{n}}$.

The $X_{1},\cdots,X_{n}$, $\frac{\partial}{\partial
y_{1}},\cdots,\frac{\partial}{\partial y_{k}}$ form a frame on
$U\times\mathbb{R}^{k}$. Because $I\left(x\right)$ depends semialgebraically
on $x\in U$, we may express

  * (15)

$I\left(x\right)=\left\\{\left(P_{1},\cdots,P_{D}\right)\in\mathcal{P}^{D}:\left.\sum_{\begin{subarray}{c}\left|\alpha\right|+\left|\beta\right|\leq
m\\\
j=1,\cdots,D\end{subarray}}\tilde{A}_{j\alpha\beta}^{i}\left(x\right)\left(X^{\alpha}\partial_{y}^{\beta}P_{j}\right)\right|_{\tilde{\psi}\left(x\right)}=0\text{,
for }i=1,\cdots,i_{\max}\right\\}$ for semialgebraic
$\tilde{A}_{j\alpha\beta}^{i}$ on $U$.

We take $U_{\text{bad}}^{1}$ to be the union of the singular sets of the
$\tilde{A}_{j\alpha\beta}^{i}$. Then $U_{\text{bad}}^{1}$ is a semialgebraic
set of dimension $<n$ in $\mathbb{R}^{n}$, and the
$\tilde{A}_{j\alpha\beta}^{i}$ are real-analytic on $U\setminus
U_{\text{bad}}^{1}$.

We may therefore rewrite the equation in ((15)) in the form

$\left.\sum_{\begin{subarray}{c}\left|\alpha\right|+\left|\beta\right|\leq
m\\\
j=1,\cdots,D\end{subarray}}\left(X^{\alpha}\left\\{A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}P_{j}\right\\}\right)\right|_{\hat{\psi}\left(x\right)}=0\text{.}$

The $A_{j\alpha\beta}^{i}$ are Nash on $U\setminus U_{\text{bad}}^{1}$. Thus,
for any closed ball $B\subset U\setminus U_{\text{bad}}^{1}$ the following
holds. (We set $\hat{B}=\hat{\psi}\left(B\right)$.)

A given $F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$ if and only if

$\sum_{\left|\alpha\right|\leq
m}X^{\alpha}\left\\{\sum_{\left|\beta\right|\leq
m-\left|\alpha\right|}A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)\right\\}=0\text{
on }\hat{B}\text{ for all }i\text{.}$

We look for integers $s\geq 0$ for which there exist Nash functions
$A_{j\alpha\beta}^{i}$ on $U\setminus U_{\text{bad}}^{1}$ with the following
property (“Property $\prod\left(s\right)$”):

Let $B\subset U\setminus U_{\text{bad}}^{1}$ be a closed ball; set
$\hat{B}=\hat{\psi}\left(B\right)$. Then $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$ if and only if

(17) $\sum_{\left|\alpha\right|\leq
s}X^{\alpha}\left\\{\sum_{\left|\beta\right|\leq
m-\left|\alpha\right|}\sum_{j=1}^{D}A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)\right\\}=0\text{
on }\hat{B}\text{ for all }i\text{.}$

We have seen that we can achieve Property $\prod\left(m\right)$.

###### Claim 5.1

Let $s$ be the smallest possible integer $\geq 0$ for which we can achieve
Property $\prod\left(s\right)$, and let $A_{j\alpha\beta}^{i}$ be as in
Property $\prod\left(s\right)$. Then $s=0$. In other words, Property
$\prod(0)$ holds.

Proof of Claim 5.1. Assuming $s\geq 1$, we will achieve Property $\prod(s-1)$,
contradicting the fact that $s$ is as small as possible.

Fix $B\subset U\setminus U_{\text{bad}}^{1}$ a closed ball, and let
$(F_{1},\cdots,F_{D})\in
C^{m}_{loc}(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D})$ be a section
of $(I(z))_{z\in\hat{B}}$. (As always, $\hat{B}=\psi(B)$.) Fix $x_{0}\in B$
and fix a multiindex $\alpha_{0}$ with $|\alpha_{0}|=s$. For $j=1,\cdots,D$,
define functions on $\mathbb{R}^{n}\times\mathbb{R}^{k}$ by setting
$F_{j}^{\\#}(z)=\theta\cdot F_{j}(z)$ where $\theta\in
C_{0}^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{k})$ with jet
$(J_{\hat{\psi}(x_{0})}\theta)(x,y)=(x-x_{0})^{\alpha_{0}}$.

Then $(F_{1}^{\\#},\cdots,F_{D}^{\\#})\in
C^{m}_{loc}(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D})$ is a section
of $(I(z))_{z\in\hat{B}}$ because each $I(z)$ is an
$\mathcal{R}_{z}$-submodule of $\mathcal{R}_{z}^{D}$.

Applying Property $\prod(s)$ to $(F_{1}^{\\#},\cdots,F_{D}^{\\#})$, we learn
that

$\left.\sum_{\left|\beta\right|\leq
m-\left|\alpha_{0}\right|}\sum_{j=1}^{D}A_{j\alpha_{0}\beta}^{i}\left(x_{0}\right)\left(\partial_{y}^{\beta}F_{j}\right)\right|_{\hat{\psi}\left(x_{0}\right)}=0\text{
}\left(\text{all }i\right)\text{.}$

This holds for all $x_{0}$ and for all $\left|\alpha_{0}\right|=s$. Thus, if
$\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$, then

(18) $\sum_{\left|\beta\right|\leq
m-\left|\alpha\right|}\sum_{j=1}^{D}A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)=0$

on $\hat{B}$ for all $\left|\alpha\right|=s$ and for all $i$. Because the
$X_{j}$ are tangent to $\hat{B}$, it follows from (18) that

(19) $X^{\alpha}\left\\{\sum_{\left|\beta\right|\leq
m-\left|\alpha\right|}\sum_{j=1}^{D}A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)\right\\}=0$

on $\hat{B}$ for all $\left|\alpha\right|=s$ and for all $i$. From (17) and
(19), we conclude that

(20) $\sum_{\left|\alpha\right|\leq
s-1}X^{\alpha}\left\\{\sum_{\left|\beta\right|\leq
m-\left|\alpha\right|}\sum_{j=1}^{D}A_{j\alpha\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)\right\\}=0$

on $\hat{B}$ for all $i$. Thus, any section of
$\left(I\left(z\right)\right)_{z\in\hat{B}}$ satisfies (18) and (20).
Conversely, suppose $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{k}\mathbb{\times\mathbb{R}}^{k},\mathbb{R}^{D}\right)$
satisfies (18) and (20). Then, because (18) implies (19), it follows that (17)
holds, and consequently $\left(F_{1},\cdots,F_{D}\right)$ is a section of
$\left(I\left(z\right)\right)_{z\in\hat{B}}$. Thus, a given
$\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$ if and only if (18)
and (20) hold. If $s\geq 1,$ this implies that we have achieved Property
$\prod\left(s-1\right)$, contradicting the minimal character of $s$, and
establishing Claim 5.1.

We return to the proof of Lemma 5.1. Because Property $\prod(s)$ holds with
$s=0$, there exist Nash functions $A_{j\beta}^{i}$ on $U\setminus
U_{\text{bad}}^{1}$, for which the following (“Property $\prod^{\ast}$”)
holds:

Let $B\subset U\setminus U_{\text{bad}}^{1}$ be a closed ball. Set
$\hat{B}=\hat{\psi}\left(B\right)$. Then a given
$\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$ if and only if

(21) $\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)=0\text{
on }\hat{B}\text{ (all }i\text{)}.$

We fix $A_{j\beta}^{i}$ as above.

We now return to our bundle
$\mathcal{H}=\left(f^{z}+I\left(z\right)\right)_{z\in\hat{U}}$.

(We abuse notation by writing $f^{z}$ for $f^{x}$ where
$z=\hat{\psi}\left(x\right)$.)

Let $B\subset U\setminus U_{\text{bad}}^{1}$ be a closed ball, and let
$\hat{B}=\hat{\psi}\left(B\right)$. Let $\left(F_{1},\cdots,F_{D}\right)$ and
$\left(\tilde{F}_{1},\cdots,\tilde{F}_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ be
any two sections of $\mathcal{H}|_{\hat{B}}$.

Then $\left(F_{1}-\tilde{F}_{1},\cdots,F_{D}-\tilde{F}_{D}\right)$ is a
section of $\left(I\left(z\right)\right)_{z\in\hat{B}}$, and therefore by
(21), we have

(22) $\sum_{\begin{subarray}{c}\left|\beta\right|\leq m\\\
j=1,\cdots,D\end{subarray}}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)=\sum_{\begin{subarray}{c}\left|\beta\right|\leq
m\\\
j=1,\cdots,D\end{subarray}}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}\tilde{F}_{j}\left(x,y\right)\text{
on }\hat{B}\text{ for all }i\text{.}$

Moreover, given $x_{0}\in B$, we can take our section
$\left(\tilde{F}_{1},\cdots,\tilde{F}_{D}\right)$ above to satisfy

$J_{\hat{\psi}\left(x_{0}\right)}\tilde{F}_{j}=f_{j}^{x_{0}}\text{
}\left(j=1,\cdots,D\right)\text{,}$

because $\left(f_{1}^{x_{0}},\cdots,f_{D}^{x_{0}}\right)\in
H\left(\hat{\psi}\left(x_{0}\right)\right)$ and $\mathcal{H}|_{\hat{B}}$ is
Glaeser stable and has nonempty fibers. (See Theorem 2.) Therefore, (22)
implies that

(23) $\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)=G^{i}\left(x\right)\text{
}$

on $\hat{B}$ for each $i$, where

$G^{i}\left(x\right)=\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\left(\partial_{y}^{\beta}f^{x}\right)|_{\hat{\psi}\left(x\right)}\text{\quad}\left(x\in
U\setminus U_{\text{bad}}^{1}\right)\text{.}$

Clearly, $G^{i}\left(x\right)$ is a semialgebraic function on $U\setminus
U_{\text{bad}}^{1}$, and it is independent of the ball $B$ in the above
discussion.

Thus, we have seen that any section $\left(F_{1},\cdots,F_{D}\right)$ of
$\mathcal{H}|_{\hat{B}}$ must satisfy (23).

Conversely, suppose $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$
satisfies (23). Let $\left(\tilde{F}_{1},\cdots,\tilde{F}_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ be
a section of $\mathcal{H}|_{\hat{B}}$. (We know that a section exists because
$\mathcal{H}|_{\hat{B}}$ is Glaeser stable and has nonempty fibers.) We know
that $\left(\tilde{F}_{1},\cdots,\tilde{F}_{D}\right)$ satisfies (23), hence

$\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}\left[F_{j}-\tilde{F}_{j}\right]\left(x,y\right)=0$

on $\hat{B}$ for each $i$.

Recalling Property $\prod^{\ast}$, we now see that
$\left(F_{1}-\tilde{F}_{1},\cdots,F_{D}-\tilde{F}_{D}\right)$ is a section of
$\left(I\left(z\right)\right)_{z\in\hat{B}}.$ Because
$\left(\tilde{F}_{1},\cdots,\tilde{F}_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of
$\mathcal{H}|_{\hat{B}}=\left(f^{z}+I\left(z\right)\right)_{z\in\hat{B}}$, we
conclude that $\left(F_{1},\cdots,F_{D}\right)$ is a section of
$\mathcal{H}|_{\hat{B}}$. Thus, if $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$
satisfies (23), then it is a section of $\mathcal{H}|_{\hat{B}}$.

We have now seen that a given $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\mathcal{H}|_{\hat{B}}$ if and only if (23) holds.

Thus, all the conclusions of Lemma 5.1 hold, except that perhaps the $G^{i}$
are not real-analytic.

We set $U_{\text{bad}}^{2}=$union of all the singular sets of the
semialgebraic functions $G^{i}$. That’s a semialgebraic set of dimension $<n$
in $\mathbb{R}^{n}$.

We take $U_{\text{bad}}=U_{\text{bad}}^{1}\cup U_{\text{bad}}^{2}$, a
semialgebraic set of dimension $<n$ in $\mathbb{R}^{n}$.

The functions $A_{j\beta}^{i}$ and $G^{i}$ are Nash on $U\setminus
U_{\text{bad}}$.

If $B\subset U\setminus U_{\text{bad}}$ is a closed ball and
$\hat{B}=\psi\left(B\right)$, then a given $\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right)$ is
a section of $\mathcal{H}|_{\hat{B}}$ if and only if

$\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\left(\partial_{y}^{\beta}F_{j}\right)|_{\hat{\psi}\left(x\right)}=G^{i}\left(x\right)$

on $B$ for each $i$.

This completes the proof of Lemma 5.1.

###### Remark 5.1

Lemma 5.1 and its proof hold also for $k=0$. In that case, $\hat{\psi}$ is the
identity map and there are no $y$-variables, hence no $y$-derivatives in the
conclusion of Lemma 5.1.

###### Corollary 5.1

Let $\mathcal{H},U,\psi,\cdots$ be as in Lemma 5.1. Let
$\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\mathbb{R}^{n}\times\mathbb{R}^{k},\mathbb{R}^{D}\right).$
Then $\left(F_{1},\cdots,F_{D}\right)$ is a section of
$\mathcal{H}|_{\hat{U}\setminus\hat{\psi}\left(U_{\text{bad}}\right)}$ if and
only if

$\sum_{\left|\beta\right|\leq
m}\sum_{j=1}^{D}A_{j\beta}^{i}\left(x\right)\partial_{y}^{\beta}F_{j}\left(x,y\right)=G^{i}\left(x\right)$

on $\hat{U}\setminus\hat{\psi}\left(U_{\text{bad}}\right)$, for all $i$.

Proof. $U\setminus U_{\text{bad}}$ is a union of (infinitely many overlapping)
closed balls $B$. Applying Lemma 5.1 to each $B$, we obtain the desired
conclusion.

###  5.2 Gaussian Elimination with Parameters

Suppose we are given a system of linear equations

  * (24)

$X_{i}+\sum_{j>k}A_{ij}X_{j}=b_{i}$, for $i=1,\cdots,k$ with
$\left|A_{ij}\right|\leq 2^{k}$ for $i=1,\cdots k,$ $j=k+1,\cdots,M$, and

  * (26)

$\sum_{j>k}C_{ij}X_{j}=g_{i}$, for $i=k+1,\cdots,N$,

where $0\leq k\leq N,M;$ the $A_{ij}$, $C_{ij}$, $b_{i}$, $g_{i}$ are
semialgebraic functions defined on a semialgebraic set
$E\subset\mathbb{R}^{n}$; and $X_{1},\cdots,X_{M}$ are unknowns.

We say that this system is in $k$-echelon form on $E$

If $k=0$, then we have simply ((26)) for $i=1,\cdots,N$, so every system of
linear equations with coefficient matrix and right-hand sides depending
semialgebraically on $x\in E$ is in $0$-echelon form on $E$.

If also $C_{ij}\equiv 0$ on $E$ for all $i=k+1,\cdots,N$, $j=k+1,\cdots,M$,
then we say that our system of equations is in echelon form on $E$. In
particular, a system in $k$-echelon form with $k=\min\\{N,M\\}$ is in echelon
form on $E$. Suppose our system is in $k$-echelon form with $k<\min\\{N,M\\}$.
We partition $E$ as follows. Let $E_{\text{good}}=\\{x\in E:\text{All the
}C_{ij}(x)=0\\}$. For $\tilde{i}=k+1,\cdots,N$ and $\tilde{j}=k+1,\cdots,M$,
we let $\tilde{E}(\tilde{i},\tilde{j})=\\{x\in
E:|C_{\tilde{i}\tilde{j}}|=\max_{ij}|C_{ij}|>0\\}$. The $E_{\text{good}}$ and
$\tilde{E}(i,j)$ form a covering of $E$.

We enumerate the pairs $(i,j)$ in any order and then form sets $E(i,j)$ by
removing from $\tilde{E}(i,j)$ all points contained in some
$\tilde{E}(i^{\prime},j^{\prime})$ with $(i^{\prime},j^{\prime})$ preceding
$(i,j)$. Then $E_{\text{good}}$ and the $E(i,j)$ form a partition of $E$ into
semialgebraic sets. On $E_{\text{good}}$, our system is in echelon form.

On each $E(a,b)$, we will exhibit a system of linear equations in
$(k+1)$-echelon form, equivalent to the given system ((24)), ((26)). For fixed
$(a,b)$, we relabel equations and unknowns so that our system still has the
form ((24)), ((26)), but with $|C_{k+1,k+1}|=\max_{ij}|C_{ij}|>0$. Dividing
equations ((26)) by $C_{k+1,k+1}$, we may assume that

(28) $C_{k+1,k+1}=1$

and all

(29) $|C_{ij}|\leq 1.$

Note that $A_{ij},C_{ij},b_{i},g_{i}$ still depend semialgebraically on $x$.
From each equation ((24)), we subtract $A_{i(k+1)}$ times equation ((26)) with
$i=k+1$. From each equation ((26)) ($i\not=k+1$), we subtract $C_{i,k+1}$
times equation ((26)) with $i=k+1$. Thus, we obtain equations of the form

(30)
$\left[\begin{array}[]{l}X_{i}+\sum_{j>k}\tilde{A}_{ij}X_{j}=\tilde{b}_{i},\text{for
}i=1,\cdots,k\\\ X_{k+1}+\sum_{j>k+1}C_{k+1,j}X_{j}=g_{k+1},\\\ \sum_{j\geq
k+1}\tilde{C}_{ij}X_{j}=\tilde{g}_{i}\text{, for }i>k+1.\end{array}\right.$

Here, $\tilde{A}_{ij}=A_{ij}-A_{i\left(k+1\right)}C_{k+1,j}$ for
$i=1,\cdots,k$, $j\geq k+1$; and $\tilde{C}_{ij}=C_{ij}-C_{i,k+1}C_{k+1,j}$
for $i=k+2,\cdots,N$, $j>k+1$.

In particular, $\tilde{A}_{i,k+1}=A_{i,k+1}-A_{i,k+1}\cdot C_{k+1,k+1}=0$, and
$\tilde{C}_{i,k+1}=C_{i,k+1}-C_{i,k+1}\cdot C_{k+1,k+1}=0$, thanks to (28).

Also,
$\left|\tilde{A}_{ij}\right|\leq\left|A_{ij}\right|+\left|A_{i,k+1}\right|\cdot\left|C_{k+1,j}\right|\leq\left|A_{ij}\right|+\left|A_{i,k+1}\right|$
(by (29))$\leq 2^{k}+2^{k}$ (because our system ((24)), ((26)) is in
$k$-echelon form)$=2^{k+1}$. Recall that $|C_{k+1,j}|\leq 1$.

These remarks show that the system of equations (30) is in
$\left(k+1\right)$-echelon form.

We repeat this procedure, starting with a system in $0$-echelon form, and
partition $E$ more and more finely into pieces $E_{\nu}$, on each of which an
equivalent system to ((24)), ((26)) is either in echelon form, or in
$k$-echelon form for ever higher $k$. The procedure has to stop after at most
$\min\left(N,M\right)$ steps, because a system in $k$-echelon form with
$k=\min\left(N,M\right)$ is automatically in echelon form.

Thus, we have proven the following result

###### Lemma 5.2

Consider a system of linear equations

(31) $\sum_{j=1}^{M}C_{ij}\left(x\right)X_{j}=g_{i}\left(x\right)\text{
}\left(i=1,\cdots,N\right)$

where the $C_{ij}\left(x\right)$ and $g_{i}\left(x\right)$ are semialgebraic
functions defined on a semialgebraic set $E\subset\mathbb{R}^{n}$.

Then we can partition $E$ into semialgebraic sets $E_{\nu}$
$\left(\nu=1,\cdots,\nu_{\max}\right)$, for which the following holds for each
$\nu$:

There exist a permutation
$\pi:\left\\{1,\cdots,M\right\\}\rightarrow\left\\{1,\cdots,M\right\\}$ and an
integer $0\leq k\leq\min\left(N,M\right)$ such that for each $x\in E_{\nu}$,
the system (31) is equivalent to a system of the form

(32) $\left[\begin{array}[]{c}X_{\pi
i}+\sum_{j>k}\tilde{A}_{ij}\left(x\right)X_{\pi
j}=\tilde{g}_{i}\left(x\right)\text{ for }i=1,\cdots,k\\\
0=\tilde{b}_{i}\left(x\right)\text{ for
}i=k+1,\cdots,N\text{.}\end{array}\right.$

That is, for each $x\in E_{\nu}$ and each
$\left(X_{1},\cdots,X_{M}\right)\in\mathbb{C}^{M}$, (31) holds at $x$ if and
only if (32) holds at $x$. Here, the $\tilde{A}_{ij},\tilde{g}_{i},$ and
$\tilde{b}_{i}$ are semialgebraic functions on $E_{\nu}$, and
$\left|\tilde{A}_{ij}\left(x\right)\right|\leq 2^{k}$ on $E_{\nu}$.

In essence, the method for solving the system (31) is just the usual Gaussian
elimination, except that we take extra care to maintain the growth condition
$\left|\tilde{A}_{ij}\left(x\right)\right|\leq 2^{k}$.

###  5.3 What It Means to be a Section of a Semialgebraic Bundle

We work with a semialgebraic bundle $\mathcal{H}=(H(x))_{x\in\mathbb{R}^{2}}$.
Each $H(x)$ is a coset of an $\mathcal{R}_{x}$-submodule of
$(\mathcal{R}_{x})^{D}$, depending semialgebraically on $x$. Here,
$\mathcal{R}_{x}$ is the ring of the $m$-jets of functions at $x$. A function
$F=(F_{1},\cdots,F_{D})\in C^{m}_{loc}(\Omega,\mathbb{R}^{D})$
($\Omega\subset\mathbb{R}^{2}$ open) is a section of $\mathcal{H}$ if for all
$x\in\Omega$ the $m$-jet $J_{x}F$ belongs to $H(x)$. A function $F\in
C^{m}_{loc}(\Omega,\mathbb{R}^{D})$ is called a local section near $x^{0}$
($x^{0}\in\Omega$) if for some small disc $B\subset\Omega$ centered at $x^{0}$
we have $J_{x}F\in H(x)$ for all $x\in B$.

Let $\Omega=\\{(x,y)\in\mathbb{R}^{2}:0\leq y\leq x\\}$. Let
$\mathcal{H}=(H(x))_{x\in\mathbb{R}^{2}}$ be a semialgebraic bundle, with
$H((0,0))=\\{0\\}$. We assume that $\mathcal{H}$ has a section. We want a
convenient condition on functions $F\in C^{m}_{loc}(\Omega,\mathbb{R}^{D})$
that is equivalent to the assertion that $F|_{B\cap\Omega^{\text{interior}}}$
is a section of $\mathcal{H}$ for a small enough disc $B$ centered at the
origin. We achieve (approximately) that.

To do so, we partition $\Omega$ into semialgebraic open subsets of
$\mathbb{R}^{2}$, finitely many semialgebraic curves in $\mathbb{R}^{2}$, and
finitely many points. To start with, we partition $\Omega$ into the point
$(0,0)$, the arcs $\\{(x,0):x>0\\},\\{(x,x):x>0\\},$ and
$\Omega^{\text{interior}}$.

As we proceed, we will cut up each of our semialgebraic open sets into
finitely many semialgebraic open subsets, finitely many semialgebraic arcs,
and finitely many points. We won’t keep track explicitly of the arcs and
points at first; we just discard semialgebraic subsets of $\mathbb{R}^{2}$ of
dimension $\leq 1$.

We apply Lemma 5.1 in the case $k=0$ to $\Omega^{\text{interior}}$ and
$\mathcal{H}$. (See Remark 5.1.)

Thus, we obtain a semialgebraic $V_{1}\subset\Omega^{\text{interior}}$ of
dimension $\leq 1$, outside of which the following holds for some
semialgebraic functions $A_{ij}^{\\#}(x),\phi_{i}^{\\#}(x)$ for $1\leq i\leq
i_{\max},1\leq j\leq D,x\in\Omega^{\text{interior}}\setminus V_{1}$:

Let $F=(F_{1},\cdots,F_{D})$ belong to $C^{m}_{loc}(U,\mathbb{R}^{D})$ where
$U$ is a neighborhood of $x^{0}\in\Omega^{\text{interior}}\setminus V_{1}$.
Then $F$ is a local section of $\mathcal{H}$ near $x^{0}$ if and only if

  * (33)

$\sum_{j=1}^{D}A_{ij}^{\\#}(x)F_{j}(x)=\phi_{i}^{\\#}(x)$, for
$i=1,\cdots,i_{\max}$, for all $x$ in a neighborhood of $x^{0}$.

The equations ((33)) have a solution for each fixed $x$, because $\mathcal{H}$
has a section. Next, we apply Lemma 5.2 to the above system of linear
equations.

Thus, we obtain a partition of $\Omega^{\text{interior}}\setminus V_{1}$ into
semialgebraic sets $E_{\nu}^{\\#}$ ($\nu=1,\cdots,\nu_{\max}^{\\#}$), for
which we have integers $\tilde{k}_{\nu}\geq 0$, permutations
$\tilde{\pi}_{\nu}:\\{1,\cdots,D\\}\rightarrow\\{1,\cdots,D\\}$, and
semialgebraic functions $\tilde{A}_{ij}^{\nu}(x)$ ($1\leq
i\leq\tilde{k}_{\nu},\tilde{k}_{\nu}+1\leq j\leq D,x\in E_{\nu}^{\\#}$),
$\tilde{\phi}_{i}^{\nu}(x)$ such that for any $x\in E_{\nu}^{\\#}$, the system
of equations ((33)) is equivalent to

(35)
$F_{\pi_{\nu}i}\left(x\right)+\sum_{j>\tilde{k}_{\nu}}\tilde{A}_{ij}^{\nu}\left(x\right)F_{\pi_{\nu}j}\left(x\right)=\tilde{\varphi}_{i}^{\nu}\left(x\right)\text{
for }i=1,\cdots,\tilde{k}_{\nu}.$

Moreover, the $\tilde{A}_{ij}^{\nu}\left(x\right)$ are bounded. Note that the
functions $\tilde{b}_{i}$ in (32) are identically $0$ because our equations
((33)) have a solution.

Because $\mathcal{H}$ has a section, there exists
$F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(\Omega,\mathbb{R}^{D}\right)$ satisfying ((33)) for all
$x\in\Omega^{\text{interior}}\setminus V_{1}$, hence also satisfying (35) in
$E_{\nu}^{\\#}$. Consequently, the left-hand side of (35) is bounded (for
bounded $x$), and thus also the $\tilde{\varphi}_{i}^{D}\left(x\right)$ are
bounded (for bounded $x$).

Applying Theorem 5, we obtain a semialgebraic $V_{2}\subset\mathbb{R}^{2}$ of
dimension $\leq 1$, satisfying

(36)
$\left|\partial^{\alpha}\tilde{\varphi}_{i}^{\nu}\left(x\right)\right|,\left|\partial^{\alpha}\tilde{A}_{ij}^{\nu}\left(x\right)\right|\leq
C\left[\text{dist}\left(x,V_{2}\right)\right]^{-\left|\alpha\right|}\text{ for
bounded }x\text{ outside }V_{2}\text{, for }\left|\alpha\right|\leq
m+100\text{.}$

By adding $\partial\Omega$ to $V_{2}$ and removing from $V_{2}$ all points
outside $\Omega$, we may assume $V_{2}\subset\Omega$. (This operation does not
increase the distance from $V_{2}$ to any point of $\Omega$.)

Let $\hat{E}_{\nu}$ $\left(\nu=1,\cdots,\nu_{\max}\right)$ be the connected
components of the interiors of the sets $E_{\nu}^{\\#}\setminus V_{2}$
($\nu=1,\cdots,\nu_{\max}^{\\#}$).

Then $\Omega$ is partitioned into the $\hat{E}_{\nu}$ and $V_{3}$, where
$V_{3}$ is a semialgebraic subset of $\Omega$ of dimension $\leq 1$. The
$\hat{E}_{\nu}$ are pairwise disjoint open connected semialgebraic sets. Any
path in $\Omega$ that does not meet $V_{3}$ stays entirely in a single
$\hat{E}_{\nu}$. Indeed, suppose not: let $\gamma\left(t\right)\in\Omega$
$\left(t\in\left[0,1\right]\right)$ be a path starting at
$\gamma\left(0\right)\in\hat{E}_{\nu}$ not staying in $\hat{E}_{\nu}$ and not
meeting $V_{3}$. Pick $t_{\ast}=$
$\inf\left\\{t>0:\gamma\left(t\right)\not\in\hat{E}_{\nu}\right\\}$. Then
$t^{*}>0$ since $\hat{E}_{\nu}$ is open. We can’t have
$\gamma\left(t_{\ast}\right)\in\hat{E}_{\nu^{\prime}}$ with
$\nu^{\prime}\not=\nu$ else $\gamma\left(t\right)\in\hat{E}_{\nu^{\prime}}$
(and $\in\hat{E}_{\nu}$) for $t\in[t_{\ast}-\varepsilon,t_{\ast})$. We can’t
have $\gamma(t_{\ast})$ in $E_{\nu}$, since that would imply $\gamma(t)$ in
$E_{\nu}$ for all $t$ in $[t_{\ast},t_{\ast}+\varepsilon]$. Thus,
$\gamma\left(t_{\ast}\right)\in V_{3}$, contradicting the fact that $\gamma$
does not meet $V_{3}$.

Moreover, there exist integers $\hat{k}_{\nu}\geq 0$, permutations
$\hat{\pi}_{\nu}:\left\\{1,\cdots,D\right\\}\rightarrow\left\\{1,\cdots,D\right\\}$,
and semialgebraic functions $\hat{A}_{ij}^{\nu}\left(x\right)$ $\left(1\leq
i\leq\hat{k}_{\nu}\text{, }\hat{k}_{\nu}+1\leq j\leq D\right)$ and
$\hat{\varphi}_{i}^{\nu}\left(x\right)$ $\left(1\leq
i\leq\hat{k}_{\nu}\right)$ defined on $\hat{E}_{\nu}$, with the following
properties

  * (37)

$\left|\partial^{\alpha}\hat{A}_{ij}^{\nu}\left(x\right)\right|$,
$\left|\partial^{\alpha}\hat{\varphi}_{i}^{\nu}\left(x\right)\right|\leq
C\left[\text{dist}\left(x,V_{3}\right)\right]^{-\left|\alpha\right|}$ for
bounded $x\in\hat{E}_{\nu}$, $\left|\alpha\right|\leq m+100$, and

  * (39)

Let $x^{0}\in\hat{E}_{\nu}$ and let $F=\left(F_{1},\cdots,F_{D}\right)$ be
$C^{m}_{loc}$ in a neighborhood of $x^{0}$. Then $F$ is a local section of
$\mathcal{H}$ near $x^{0}$ if and only if

$F_{\pi_{\nu}i}\left(x\right)+\sum_{j>\hat{k}_{\nu}}\hat{A}_{ij}^{\nu}\left(x\right)F_{\pi_{\nu}j}\left(x\right)=\hat{\varphi}_{i}^{\nu}\left(x\right)$

in a neighborhood of $x^{0}$ for each $i=1,\cdots,\hat{k}_{\nu}$.

We partition $V_{3}\cup\left\\{\left(x,0\right):x\geq
0\right\\}\cup\left\\{\left(x,x\right):x\geq 0\right\\}$ into finitely many
Nash open arcs (not containing their endpoints) and finitely many points.

For small enough $\delta>0$,
$B\left(0,\delta\right)\mathbb{\subset\mathbb{R}}^{2}$ avoids all the above
arcs not containing $0$ in their closure, and all the above points except
possibly for the point $0$. Taking $\delta$ small, we may assume that the
remaining arcs have convergent Puiseux series in $B(0,\delta)$.

Notice that our semialgebraic one-dimensional sets are all contained in
$\Omega$; so no arcs have tangent lines at $0$ lying outside the sector
$\Omega$. Thus, the remaining arcs have the form $\\{y=\psi_{s}(x)\\}$ in
$B(0,\delta)$, where $\psi_{1},\cdots,\psi_{s_{\max}}$ are semialgebraic
functions of one variable, with convergent Puiseux expansion in $[0,\delta]$.
We discard duplicates, i.e., we may assume $\psi_{s}$ is never identically
equal to $\psi_{s^{\prime}}$ for $s^{\prime}\not=s$. Note that the line
segments $\\{(x,0):0<x<\delta\\}$ and $\\{(x,x):0<x<\delta\\}$ are among our
arcs $\gamma_{s}$. Taking $\delta>0$ smaller yet, we may assume that for each
$s\not=s^{\prime}$, either $\psi_{s}(x)<\psi_{s^{\prime}}(x)$ for all
$x\in(0,\delta)$, or $\psi_{s}(x)>\psi_{s^{\prime}}(x)$ for all
$x\in(0,\delta)$. (That’s because the $\psi_{s}$ are given by convergent
Puiseux expansions.) Thus, in $B(0,\delta)$, our curves may be labelled so
that $0\equiv\psi_{0}(x)<\psi_{1}(x)<\cdots<\psi_{s_{\max}}(x)\equiv x$ for
$x\in(0,\delta)$. The arcs are
$\gamma_{s}=\\{(x,\psi_{s}(x)):x\in[0,\delta]\\}$ for $s=0,\cdots,s_{\max}$.
(Here we have thrown in the point $0$, and taken $\delta$ small to allow
ourselves to include $x=\delta$, not just $x<\delta$.)

The sets we discarded in passing from $V_{3}$ to the semialgebraic arcs
$\gamma_{0},\cdots,\gamma_{s_{\max}}$ are irrelevant in the sense that
$V_{3}\cap
B(0,\delta)\subset(\gamma_{0}\cup\gamma_{1}\cup\cdots\cup\gamma_{s_{\max}})\cap
B(0,\delta)$.

Let $E_{s}$ ($s=1,\cdots,s_{\max}$) be the part of the $B(0,\delta)$ lying
between $\gamma_{s-1}$ and $\gamma_{s}$, i.e., $E_{s}=\\{(x,y)\in
B(0,\delta):0<x<\delta,\psi_{s-1}(x)<y<\psi_{s}(x)\\}$.

Any two points in a given $E_{s}$ may be joined by a path in
$B(0,\delta)\setminus\bigcup_{s=0}^{s_{\max}}\gamma_{s}\subset
B(0,\delta)\setminus V_{3}$, hence all points in a given $E_{s}$ lie in the
same $\hat{E}_{\nu}$.

Therefore, for $s=1,\cdots,s_{\max}$, there exist $k_{s}\geq 0$, permutations
$\pi_{s}:\\{1,\cdots,D\\}\rightarrow\\{1,\cdots,D\\}$, and semialgebraic
functions $A_{ij}^{s}(x)$, $\psi_{i}^{s}(x)$ ($1\leq i\leq
k_{s};j=k_{s}+1,\cdots,D$) on $E_{s}$, with the following properties

  * (41)

Let $x^{0}\in E_{s}$, and let $F=(F_{1},\cdots,F_{D})$ be $C^{m}_{loc}$ in a
neighborhood of $x^{0}$. Then $F$ is a local section of $\mathcal{H}$ near
$x^{0}$ if and only if

  * (43)

$F_{\pi_{s}i}(x)+\sum_{j>k_{s}}A_{ij}^{s}(x)F_{\pi_{s}j}(x)=\psi_{i}^{s}(x)$
in a neighborhood of $x^{0}$ for each $i=1,\cdots,k_{s}$.

Moreover,

  * (45)

$|\partial^{\alpha}A_{ij}^{s}(x)|$, $|\partial^{\alpha}\psi_{i}^{s}(x)|\leq
C\left[\text{dist}(x,\gamma_{s}\cup\gamma_{s-1})\right]^{-|\alpha|}$ on
$E_{s}$ for $|\alpha|\leq m+100$.

In particular, if $F=(F_{1},\cdots,F_{D})\in
C^{m}_{loc}(\Omega,\mathbb{R}^{D})$, then $J_{x}F\in H(x)$ for all
$x\in[\Omega\cap
B(0,\delta)]\setminus(\gamma_{0}\cup\cdots\cup\gamma_{s_{\max}})$ if and only
if for each $s=1,\cdots,s_{\max}$, ((43)) holds on all of $E_{s}$.

Next, we apply Lemma 5.1 to $\mathcal{H}_{s}=(H(x))_{x\in\gamma_{s}}$,
($s=0,\cdots,s_{\max}$). We obtain semialgebraic functions for which the
following holds.

Let $\left(x^{0},\psi_{s}\left(x^{0}\right)\right)\in\gamma_{s}$ be given, and
let $F=\left(F_{1},\cdots,F_{D}\right)\in
C^{m}_{loc}\left(U,\mathbb{R}^{D}\right)$, where $U$ is a neighborhood of
$\gamma_{s}$ in $\mathbb{R}^{2}$. Then, except for finitely many bad $x^{0}$,
we have the following equivalence:

$F$ is a local section of $\mathcal{H}_{s}$ near
$\left(x^{0},\psi_{s}\left(x^{0}\right)\right)$ if and only if

$\sum_{\begin{subarray}{c}1\leq j\leq D\\\ 0\leq l\leq
m\end{subarray}}\Theta_{jl}^{is}\left(x\right)\partial_{y}^{l}F_{j}|_{\left(x,\psi_{s}\left(x\right)\right)}=g^{si}\left(x\right)\quad\left(i=1,\cdots,i_{\max}\left(s\right)\right)$

for all $x$ in a neighborhood of $x^{0}$. Here, the $\Theta$’s and $g$’s are
semialgebraic functions of one variable. To say that $F$ is a local section of
$\mathcal{H}_{s}$ near $\left(x^{0},\psi_{s}\left(x^{0}\right)\right)$ means
that $J_{\left(x,\psi_{s}\left(x\right)\right)}F\in
H\left(x,\psi_{s}\left(x\right)\right)$ for all $x$ in a neighborhood of
$x^{0}$.

By restricting attention to $B\left(0,\delta\right)$ and taking $\delta>0$
smaller, we may exclude from $B\left(0,\delta\right)$ all these bad $x^{0}$,
except for $x^{0}=0$.

Combining our results ((41)), ((45)) on the $E_{\nu}$ with the above result on
the arcs $\gamma_{s}$, we obtain the following result.

###### Lemma 5.3

Let $\Omega=\left\\{\left(x,y\right)\in\mathbb{R}^{2}:0\leq y\leq x\leq
1\right\\}$ and let $\mathcal{H=}\left(H\left(x\right)\right)_{x\in\Omega}$ be
a semialgebraic bundle, with each $H\left(x\right)$ consisting of $m$-jets at
$x$ of functions from $\mathbb{R}^{2}$ to $\mathbb{R}^{D}$.

Assume $H\left(\left(0,0\right)\right)=\left\\{0\right\\}$ and assume
$\mathcal{H}$ has a section.

Then there exist the following objects, with properties to be specified below:

  * •

A positive number $\delta\in\left(0,1\right)$.

  * •

Semialgebraic functions
$0=\psi_{0}\left(x\right)<\psi_{1}\left(x\right)<\cdots<\psi_{s_{\max}}\left(x\right)=x$
on $\left(0,\delta\right),$ all given by convergent Puiseux expansions on
$\left(0,\delta\right)$.

  * •

Integers $k_{s}$ $\left(0\leq k_{s}\leq D\right)$ and permutations
$\pi_{s}:\left\\{1,\cdots,D\right\\}\rightarrow\left\\{1,\cdots,D\right\\}$
for $s=1,\cdots,D$.

  * •

Semialgebraic functions $A_{ij}^{s}\left(x,y\right)$
$\left(s=1,\cdots,s_{\max},1\leq i\leq k_{s},k_{s}<j\leq D\right)$ and
$\varphi_{i}^{s}\left(x,y\right)$ $(s=1,\cdots,s_{\max},1\leq i\leq k_{s})$
defined on
$E_{s}=\left\\{\left(x,y\right):0<x<\delta,\psi_{s-1}\left(x\right)<y<\psi_{s}\left(x\right)\right\\}$.

  * •

Semialgebraic functions $\Theta_{jl}^{si}\left(x\right)$,
$g^{si}\left(x\right)$
$(s=0,\cdots,s_{\max},i=1,\cdots,i_{\max}\left(s\right)$, $j=1,\cdots,D,$
$l=0,\cdots,m$ defined on $\left(0,\delta\right)$, and given there by there by
convergent Puiseux expansions.

The above objects have the following properties

  * •

(Estimates) For $\left(x,y\right)\in\Omega$ with $0<x<\delta$ and
$\psi_{s-1}\left(x\right)<y<\psi_{s}\left(x\right)$, we have
$\left|\partial^{\alpha}A_{ij}^{s}\left(x,y\right)\right|$,
$\left|\partial^{\alpha}\varphi_{i}^{s}\left(x,y\right)\right|\leq
C\left[\min\left(\left|y-\psi_{s}\left(x\right)\right|,\left|y-\psi_{s-1}\left(x\right)\right|\right)\right]^{-\left|\alpha\right|}$
for $\left|\alpha\right|\leq m+100$.

  * •

(Condition for sections) Let $F=(F_{1},...,F_{D})\in
C^{m}_{loc}(\Omega,\mathbb{R}^{D})$, and suppose $J_{x}F\in H\left(x\right)$
for all $x\in\Omega$.

    * (47)

Then for $s=1,\cdots,s_{\max}$, $i=1,\cdots,k_{s}$,
$x\in\left(0,\delta\right)$,
$\psi_{s-1}\left(x\right)<y<\psi_{s}\left(x\right)$, we have

$F_{\pi_{s}i}\left(x,y\right)+\sum_{D\geq
j>k_{s}}A_{ij}^{s}\left(x,y\right)F_{\pi_{s}j}\left(x,y\right)=\varphi_{i}^{s}\left(x,y\right)\text{;}$

and for $s=0,1,\cdots,s_{\max}$, $i=1,\cdots,i_{\max}\left(s\right)$,
$x\in\left(0,\delta\right)$, we have

$\sum_{j=1}^{D}\sum_{l=0}^{m}\Theta_{jl}^{si}\left(x\right)\partial_{y}^{l}F_{j}\left(x,\psi_{s}\left(x\right)\right)=g^{si}\left(x\right)\text{;}$

and $J_{\left(0,0\right)}F_{j}=0$ for all $j$.

Conversely, if $F=(F_{1},...,F_{D})\in C^{m}_{loc}(\Omega,\mathbb{R}^{D})$ and
the conditions in ((47)) are satisfied, then $J_{z}F$ $\in H\left(z\right)$
for all $z=\left(x,y\right)\in\Omega$ with $0\leq x<\delta$.

##  6 A Second Main Lemma

This section is devoted to the proof of the following lemma. See (A) and (B)
in the Introduction.

######  Lemma 6.1 (Second Main Lemma)

Let $\mathcal{H}=(H(z))_{z\in\Omega}$ with
$\Omega=\left\\{(x,y)\in\mathbb{R}^{2}:0\leq y\leq x\leq 1\right\\}$ and
suppose $H(z)$ depends semialgebraically on $z$. (As usual,
$H(z)\subset\mathcal{R}_{z}^{D}$ is a coset of an
$\mathcal{R}_{z}$-submodule.)

Suppose $\mathcal{H}$ has a section, and suppose
$\mathcal{H}((0,0))=\left\\{0\right\\}$. Then there exist semialgebraic
functions $\theta_{jl}^{si}(x)$, $g^{si}(x)$, $\tilde{\theta}_{jl}^{si}(x)$,
$\tilde{g}^{si}(x)$ of one variable, and
$0=\psi_{0}(x)<\cdots<\psi_{s_{\max}}(x)=x$, also semialgebraic, for which the
following hold.

Suppose $F=(F_{1},\cdots,F_{D})\in C^{m}(\Omega,\mathbb{R}^{D})$ is a section
of $\mathcal{H}$. Let $f_{jl}^{s}(x)=\partial_{y}^{l}F_{j}(x,\psi_{s}(x))$ for
$0\leq s\leq s_{\max}$, $0\leq l\leq m$, $1\leq j\leq D$.

Then

  1. (49)

$\sum_{j,l}\theta_{jl}^{si}(x)f_{jl}^{s}(x)=g^{si}(x)$ on $(0,\delta)$ for
some $\delta>0$ for each $s,i$; and
$\sum_{j,l}\tilde{\theta}_{jl}^{si}(x)f_{jl}^{s}(x)=\tilde{g}^{si}(x)+o(1)$ as
$x\rightarrow 0^{+}$, each $s$, $i$; and
$f_{jl}^{s}(x)=\sum_{k=0}^{m-l}\frac{1}{k!}f_{j(l+k)}^{s-1}(x)\cdot\left(\psi_{s}\left(x\right)-\psi_{s-1}\left(x\right)\right)^{k}+o\left(\left[\psi_{s}\left(x\right)-\psi_{s-1}\left(x\right)\right]^{m-l}\right)$
as $x\rightarrow 0^{+}$, each $s$, $j$, $l$.

  1. (51)

Conversely, if $f_{jl}^{s}\left(x\right)$ are semialgebraic functions
satisfying ((49)), then there exists a semialgebraic $C^{m}$ section
$F=\left(F_{1},\cdots,F_{D}\right)$ of $\mathcal{H}$ over
$\Omega_{\delta^{\prime}}=\left\\{\left(x,y\right):0\leq y\leq
x\leq\delta^{\prime}\right\\}$ (some $\delta^{\prime}>0$) such that
$\partial_{y}^{l}F_{j}\left(x,\psi_{s}\left(x\right)\right)=f_{jl}^{s}\left(x\right)$
for $0<x<\delta^{\prime}$.

We call the curves $y=\psi_{s}(x)$ “critical curves”.

###  6.1 The Jet of a Section at a Critical Curve

Fix $m\geq 1$. Recall that $\mathcal{P}$ denotes the space of polynomials of
degree $\leq m$ on $\mathbb{R}^{2}$, and $J_{z}F\in\mathcal{P}$ denotes the
$m$-jet of $F$ at $z\in\mathbb{R}^{2}$. $\odot_{z}$ denotes multiplication of
jets at $z$. We write $\mathfrak{p}$ to denote the space of polynomials of
degree $\leq m$ on $\mathbb{R}$. If $F(x,y)$ is a $C^{m}_{loc}$ function in a
neighborhood of $(\bar{x},0)$, then $j_{\bar{x}}F\in\mathfrak{p}$ is the
$m$-jet at $0$ of the function $y\mapsto F(\bar{x},y)$. We write $\boxdot$ to
denote multiplication of $m$-jets at $0$ of $C^{m}_{loc}$ functions of one
variable.

If $\vec{F}=(F_{1},\cdots,F_{j_{\max}})$ is a vector of $C^{m}_{loc}$
functions on $\mathbb{R}^{2}$, then $J_{z}\vec{F}$ denotes

$(J_{z}F_{1},\cdots,J_{z}F_{j_{\max}})\in\mathcal{P}^{j_{\max}}.$

Similarly, $j_{\bar{x}}\vec{F}$ denotes
$(j_{\bar{x}}F_{1},\cdots,j_{\bar{x}}F_{j_{\max}})\in\mathfrak{p}^{j_{\max}}.$

A function $F^{\\#}:(0,\delta)\rightarrow\mathfrak{p}$ may be regarded as a
function of $(x,y)\in(0,\delta)\times\mathbb{R}$ such that for fixed $x$, the
function $y\mapsto F^{\\#}(x,y)$ is a polynomial of degree at most $m$.

Fix positive integers $i_{\max},j_{\max}$. Let Aff denote the vector space of
all affine functions defined on $\mathfrak{p}^{j_{\max}+i_{\max}}$. We make
the following assumptions:

  * •

We are given $C^{\infty}$ semialgebraic functions
$A_{ij},B_{i},(i=1,\cdots,i_{\max},j=1,\cdots,j_{\max})$ defined on
$\Omega_{1}$, where for $\delta>0$,
$\Omega_{\delta}=\\{(x,y)\in\mathbb{R}^{2}:0<x<\delta,0<y<\psi(x)\\}$, and
$\psi:(0,1)\rightarrow(0,\infty)$ is a semialgebraic function satisfying
$0<\psi(x)\leq x$ for $x\in(0,1)$.

  * •

We assume that $\partial^{\alpha}A_{ij},\partial^{\alpha}B_{i}$ extend to
continuous functions on $\Omega_{1}^{+}$ for $|\alpha|\leq m$, where, for
$\delta>0$,
$\Omega_{\delta}^{+}=\\{(x,y)\in\mathbb{R}^{2}:0<x\leq\delta,0<y\leq\psi(x)\\}$.

  * •

We suppose that

$\displaystyle|\partial^{\alpha}A_{ij}(x,y)|$ $\displaystyle\leq$
$\displaystyle Cy^{-|\alpha|},\text{ and}$
$\displaystyle|\partial^{\alpha}B_{i}(x,y)|$ $\displaystyle\leq$
$\displaystyle Cy^{-|\alpha|}$

on $\Omega^{+}_{1}$ for $|\alpha|\leq m$.

###### Lemma 6.2

Under the above assumptions, there exist $\delta\in(0,1)$ and semialgebraic
maps
$\lambda_{1},\cdots,\lambda_{k_{\max}},\mu_{1},\cdots,\mu_{l_{\max}}:(0,\delta)\rightarrow\text{Aff}$
such that the following hold:

  * (53)

Suppose $\vec{F}=(F_{1},\cdots,F_{j_{\max}})$ and
$\vec{G}=(G_{1},\cdots,G_{i_{\max}})$ belong to
$C^{m}(\Omega_{\delta}^{\text{closure}},\mathbb{R}^{j_{\max}})$ and
$C^{m}(\Omega_{\delta}^{\text{closure}},\mathbb{R}^{i_{\max}})$ respectively,
with $J_{(0,0)}\vec{F}=0,J_{(0,0)}\vec{G}=0$. Suppose also that
$G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$ for each $i$. Then
$[\lambda_{k}(\bar{x})](j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G})=0$ for
$k=1,\cdots,k_{\max},\bar{x}\in(0,\delta)$, and
$[\mu_{l}(\bar{x})](j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G})$ is bounded on
$(0,\delta)$ and tends to zero as $\bar{x}\rightarrow 0$, for each
$l=1,\cdots,l_{\max}$. We do not assume $\vec{F}$ or $\vec{G}$ is
semialgebraic.

  * (55)

Suppose there exists an $(\vec{F},\vec{G})$ as in ((53)). Let
$\vec{F}^{\\#}=(F_{1}^{\\#},\cdots,F_{j_{\max}}^{\\#})$,
$\vec{G}^{\\#}=(G_{1}^{\\#},\cdots,G_{i_{{}_{\max}}}^{\\#})$, where the
$F_{j}^{\\#}$ and $G_{i}^{\\#}$ are semialgebraic maps from
$(0,\delta)\rightarrow\mathfrak{p}$. Suppose that

$[\lambda_{k}(\bar{x})](\vec{F}^{\\#}(\bar{x}),\vec{G}^{\\#}(\bar{x}))=0,$

for $k=1,\cdots,k_{\max},\bar{x}\in(0,\delta)$; and that
$[\mu_{l}(\bar{x})](\vec{F}^{\\#}(\bar{x}),\vec{G}^{\\#}(\bar{x}))$ is bounded
on $(0,\delta)$ and tends to zero as $\bar{x}\rightarrow 0$. Then there exist
$\delta^{\prime}>0$ and $\vec{F}=(F_{1},\cdots,F_{j_{\max}})$,
$\vec{G}=(G_{1},\cdots,G_{i_{\max}})$ semialgebraic and in
$C^{m}(\Omega_{\delta^{\prime}}^{\text{closure}},\mathbb{R}^{j_{\max}})$ and
$C^{m}(\Omega_{\delta^{\prime}}^{\text{closure}},\mathbb{R}^{i_{\max}})$
respectively, with $J_{(0,0)}\vec{F}=0,J_{(0,0)}\vec{G}=0$,
$G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$ and
$j_{\bar{x}}\vec{F}=\vec{F}^{\\#}(\bar{x}),j_{\bar{x}}\vec{G}=\vec{G}^{\\#}(\bar{x})$,
for all $\bar{x}\in(0,\delta^{\prime})$. (Note that here we have passed from
$\delta$ to a smaller $\delta^{\prime}$.)

The remainder of this section is devoted to a proof of Lemma 6.2.

Let $\delta>0$ be small enough to be picked below,

###### Definition 6.1

We define a bundle $\mathcal{H}$ over
$[0,1]\times\\{0\\}\subset\mathbb{R}^{2}$. Here,
$\mathcal{H}=(H(\bar{x},0))_{\bar{x}\in[0,1]}$, with
$H(\bar{x},0)\subset\mathcal{P}^{j_{\max}+i_{\max}}$ defined as follows.

  * •

$H(0,0)=\\{0\\}$.

  * •

If $\bar{x}\in(0,1]$, then
$(\vec{P},\vec{Q})=(P_{1},\cdots,P_{j_{\max}},Q_{1},\cdots,Q_{i_{\max}})\in
H(\bar{x},0)$ if and only if

$y^{|\alpha|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{i}-Q_{i}\right\\}(\bar{x},y)\rightarrow
0$

as $y\rightarrow 0^{+}$, for each $|\alpha|\leq m$ and each $i$.

We will show that $\mathcal{H}$ is a bundle, i.e., $H(z)$ is a translate of an
$\mathcal{R}_{z}$-submodule of $\mathcal{R}_{z}^{j_{\max}+i_{\max}}$ for each
$z\in[0,\delta]\times\\{0\\}$; and we will show that
$J_{(\bar{x},0)}(\vec{F},\vec{G})\in H(\bar{x},0)$ (each
$\bar{x}\in[0,\delta]$) if $\vec{F},\vec{G}$ are as in ((53)).

Suppose $J_{(0,0)}(\vec{F},\vec{G})=0$, $\vec{F},\vec{G}$ are $C^{m}$ on
$\Omega_{\delta}^{\text{closure}}$, $G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$ on
$\Omega_{\delta}$. Let $\bar{x}\in(0,\delta]$. Then

$\partial^{\alpha}[A_{ij}(F_{j}-J_{(\bar{x},0)}F_{j})](\bar{x},y)=o(y^{m-|\alpha|})$

and

$\partial^{\alpha}[G_{i}-J_{(\bar{x},0)}G_{i}](\bar{x},y)=o(y^{m-|\alpha|})$

on $\Omega_{\delta}$ for $|\alpha|\leq m$, by Taylor’s theorem and our
estimates for $\partial^{\alpha}A_{ij}$. The above remarks imply that
$\partial^{\alpha}\\{\sum_{j}A_{ij}J_{(\bar{x},0)}F_{j}+B_{i}-J_{(\bar{x},0)}G_{i}\\}(\bar{x},0)=o(y^{m-|\alpha|})$.

Therefore, $J_{(\bar{x},0)}(\vec{F},\vec{G})\in H(\bar{x},0)$ for
$\bar{x}\in(0,\delta]$. For $\bar{x}=0$, we just note that
$J_{(0,0)}(\vec{F},\vec{G})=0\in H(0,0)$. That proves our assertion about
$J_{(\bar{x},0)}(\vec{F},\vec{G})$.

Note that for $\bar{x}\not=0,$ $H\left(\bar{x},0\right)$ is a translate in
$\mathcal{P}$ of

$I\left(\bar{x}\right)=\left\\{\left(\vec{P},\vec{Q}\right):\partial^{\alpha}\left(\sum_{j}A_{ij}P_{i}-Q_{i}\right)\left(\bar{x},y\right)=o\left(y^{m-\left|\alpha\right|}\right)\text{,
as }y\rightarrow 0^{+}\text{, }\left|\alpha\right|\leq m\right\\}\text{.}$

Let $\left(\vec{P},\vec{Q}\right)\in I\left(\bar{x}\right)$ and let
$S\in\mathcal{P}$. Then for $\left|\alpha\right|\leq m,$ we have

$\partial^{\alpha}\left(S\cdot\left[\sum_{j}A_{ij}P_{j}-Q_{i}\right]\right)\left(\bar{x},y\right)=o\left(y^{m-\left|\alpha\right|}\right),$

hence

(57)
$\partial^{\alpha}\left(\sum_{j}A_{ij}\left(SP_{j}\right)-\left(SQ_{i}\right)\right)\left(\bar{x},y\right)=o\left(y^{m-\left|\alpha\right|}\right)\text{,
as }y\rightarrow 0^{+}\text{.}$

Also, our estimates on $\partial^{\alpha}A_{ij},$ together with Taylor’s
theorem, give

$\partial^{\alpha}\left(A_{ij}\left(SP_{i}-J_{\left(\bar{x},0\right)}\left(SP_{j}\right)\right)\right)\left(\bar{x},0\right)=o\left(y^{m-\left|\alpha\right|}\right)$

and

$\partial^{\alpha}\left(SQ_{i}-J_{\left(\bar{x},0\right)}\left(SQ_{i}\right)\right)\left(\bar{x},0\right)=o\left(y^{m-\left|\alpha\right|}\right)\text{
as }y\rightarrow 0^{+}\text{ for }\left|\alpha\right|\leq m\text{.}$

That is,

(58)
$\partial^{\alpha}\left(A_{ij}\left(SP_{j}-S\odot_{\left(\bar{x},0\right)}P_{j}\right)\right)\left(\bar{x},y\right)=o\left(y^{m-\left|\alpha\right|}\right)$

and

(59)
$\partial^{\alpha}\left(SQ_{i}-S\odot_{\left(\bar{x},0\right)}Q_{i}\right)\left(\bar{x},0\right)=o\left(y^{m-\left|\alpha\right|}\right)\text{
as }y\rightarrow 0^{+}\text{ for }\left|\alpha\right|\leq m.$

It now follows from (57), (58), and (59) that

$\partial^{\alpha}\left(\sum_{j}A_{ij}\left[S\odot_{\left(\bar{x},0\right)}P_{j}\right]-\left[S\odot_{\left(\bar{x},0\right)}Q_{i}\right]\right)\left(\bar{x},y\right)=o\left(y^{m-\left|\alpha\right|}\right)$

as $y\rightarrow 0^{+}$, for each $\left|\alpha\right|\leq m$.

This completes the proof that the $I\left(\bar{x}\right)$ is a submodule, when
$\bar{x}\not=0$.

For $\bar{x}=0,$ we just note that $\left\\{0\right\\}$ is an
$\mathcal{R}_{\left(0,0\right)}$-submodule of
$\mathcal{R}_{\left(0,0\right)}^{j_{\max}+i_{\max}}.$

We have now shown that

  * •

$\mathcal{H}=(H(\bar{x},0))_{\bar{x}\in[0,\delta]}$ is a bundle.

  * •

If $(\vec{F},\vec{G})$ is as in (I) of Lemma 6.2, then $(\vec{F},\vec{G})$ is
a section of $\mathcal{H}$.

  * •

$H(\bar{x},0)\subset\mathcal{P}^{j_{\max}+i_{\max}}$ depends semialgebraically
on $\bar{x}$, since $A_{ij}$ and $B_{i}$ are semialgebraic.

###### Lemma 6.3

Let $\mathcal{H}=(H(\bar{x},0))_{(\bar{x},0)\in[0,\delta]\times\\{0\\}}$ be a
semialgebraic bundle,
$\mathcal{H}=H(\bar{x},0)\subset\mathcal{P}^{j_{\max}+i_{\max}}$. Then there
exist semialgebraic functions
$\lambda_{1},\cdots,\lambda_{k_{\max}}:(0,\delta)\rightarrow\text{Aff}$, and a
finite set of bad points
$\\{\bar{\bar{x}}_{1}^{\text{bad}},\cdots,\bar{\bar{x}}_{S}^{\text{bad}}\\}$
such that the following holds for any $\bar{\bar{x}}\in\left(0,\delta\right)$
other than the bad points. Let
$\left(\vec{F},\vec{G}\right)=\left(F_{1},\cdots,F_{j_{\max}},G_{1},\cdots,G_{i_{\max}}\right)$
be $C^{m}$ in a neighborhood of $\left(\bar{\bar{x}},0\right)$ in
$\mathbb{R}^{2}$. Then

$J_{\left(\bar{x},0\right)}\left(\vec{F},\vec{G}\right)\in
H\left(\bar{x},0\right)\text{ for all }\bar{x}\text{ in some neighborhood of
}\bar{\bar{x}}$

if and only if

$\left[\lambda_{k}\left(\bar{x}\right)\right]\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G}\right)=0\text{
for all }\bar{x}\text{ in some neighborhood of
}\bar{\bar{x}},(k=1,\cdots,k_{\max})\text{.}$

Proof. This is a 1 dimensional case of Lemma 5.1, whose proof can be found in
Section 5.1.

Proof of Lemma 6.2. We apply Lemma 6.3 to the bundle $\mathcal{H}$ defined in
Definition 6.1. By making $\delta$ smaller, we may assume there are no bad
points $\bar{\bar{x}}_{{}^{\text{bad}}}$. Thus, we have achieved the
following: There exist semialgebraic functions
$\lambda_{1},\cdots,\lambda_{k_{\max}}:(0,\delta]\rightarrow\text{Aff}$ such
that for any $\bar{\bar{x}}\in(0,\delta)$ and any $(\vec{F},\vec{G})$ that is
$C^{m}$ in a neighborhood of $(\bar{\bar{x}},0)$, we have

$J_{\left(\bar{x},0\right)}(\vec{F},\vec{G})\in H(\bar{x},0)\text{ for all
}\bar{x}\text{ in some neighborhood of }\bar{\bar{x}}$

if and only if

$\left[\lambda_{k}\left(\bar{x}\right)\right]j_{\bar{x}}\left(\vec{F},\vec{G}\right)=0\text{
for all }\bar{x}\text{ in some neighborhood of
}\bar{\bar{x}},(k=1,\cdots,k_{\max})\text{.}$

In particular, if $\left(\vec{F},\vec{G}\right)$ is as in ((53)), then

$\left[\lambda_{k}\left(\bar{x}\right)\right]j_{\bar{x}}\left(\vec{F},\vec{G}\right)=0\text{
for all }\bar{x}\in(0,\delta),(k=1,\cdots,k_{\max})\text{.}$

Next, we apply Theorem 6 in Section 3.7.

Recall $H(\bar{x},0)$ is an affine space, so $\mathbb{R}\cdot H(\bar{x},0)$ is
a vector space.

We regard $\mathbb{R}\cdot H(\bar{x},0)$ as the space of all
$(\vec{P},\vec{Q},t)$ such that
$\partial^{\alpha}\\{\sum_{j}A_{ij}P_{j}+tB_{i}-Q_{i}\\}(\bar{x},y)=o(y^{m-|\alpha|})$
as $y\rightarrow 0^{+}$.

We define seminorms on $\mathbb{R}\cdot H(\bar{x},0)$ by

$|||(\vec{P},\vec{Q},t)|||_{\alpha,i,y}=\left|y^{|\alpha|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+tB_{i}-Q_{i}\right\\}(\bar{x},y)\right|$

for fixed $\bar{x}$ and $0<y<\psi(\bar{x})$. Notice that on $H(\bar{x},0)$,
the seminorm agrees with

$|||(\vec{P},\vec{Q})|||_{\alpha,i,y}=\left|y^{|\alpha|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{i}-Q_{i}\right\\}(\bar{x},y)\right|$

for fixed $\bar{x}\not=0$ and $0<y<\psi(\bar{x}),|\alpha|\leq
m,i=1,\cdots,i_{\max}$.

Note that

$\sup_{\alpha,i,y}|||(\vec{P},\vec{Q})|||_{\alpha,i,y}$

is bounded for fixed $\left(\vec{P},\vec{Q}\right)\in
H\left(\bar{x},0\right)$, by definition of $H\left(\bar{x},0\right)$.

Thus, by Theorem 6 in Section 3.7, for each $\bar{x}\in\left(0,\delta\right),$
there exist $y_{\sigma}\in\left(0,\psi\left(\bar{x}\right)\right)$
$\left(\sigma=1,\cdots,\sigma_{\max}\right)$ with $\sigma_{\max}$ depending
only on $i_{\max},j_{\max},m$ such that for any
$\left(\vec{P},\vec{Q}\right)\in H\left(\bar{x},0\right)$, we have

(60) $\displaystyle\sup_{\begin{subarray}{c}0<y<\psi\left(\bar{x}\right)\\\
\left|\alpha\right|\leq m\\\
i=1,\cdots,i_{\max}\end{subarray}}\left|y\right|^{\left|\alpha\right|-m}\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{j}-Q_{i}\right\\}\left(\bar{x},y\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\max_{\begin{subarray}{c}\sigma=1,\cdots,\sigma_{\max}\\\
\left|\alpha\right|\leq m\\\
i=1,\cdots,i_{\max}\end{subarray}}\left|y_{\sigma}\right|^{\left|\alpha\right|-m}\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{j}-Q_{i}\right\\}\left(\bar{x},y_{\sigma}\right)\right|\text{.}$

Moreover, (60) is a semialgebraic condition. Therefore, we may take
$y_{1},\cdots,y_{\sigma}\in\left(0,\psi\left(\bar{x}\right)\right)$ satisfying
(60) to depend semialgebraically on $\bar{x}\in\left(0,\delta\right)$.

Because $0<y_{\sigma}\left(\bar{x}\right)<\psi\left(\bar{x}\right)\leq\bar{x}$
for $\bar{x}\in\left(0,\delta\right)$ and because
$y_{\sigma}\left(\bar{x}\right)$ depends semialgebraically on $\bar{x}$, we
can take $\delta$ small to achieve the estimates

  * (61)

$\left|\left(\frac{d}{dx}\right)^{\alpha}y_{\sigma}\left(\bar{x}\right)\right|\leq
C\bar{x}^{1-\alpha}$ for $0\leq\alpha\leq m+100$,
$\sigma=1,\cdots,\sigma_{\max},$ $\bar{x}\in\left(0,\delta\right).$

  * (63)

$0<y_{\sigma}(\bar{x})<\psi(\bar{x})\leq\bar{x}$ for
$\sigma=1,\cdots,\sigma_{\max}$, $\bar{x}\in\left(0,\delta\right)$.

  * (65)

$\bar{x}\mapsto y_{\sigma}(\bar{x})$ is a semialgebraic function.

  * (67)

For any $\bar{x}\in(0,\delta)$ and any
$\left(\vec{P},\vec{Q}\right)=\left(P_{1},\cdots,P_{j_{\max}},Q_{1},\cdots,Q_{i_{\max}}\right)\in
H\left(\bar{x},0\right),$ we have

$\displaystyle\sup_{\begin{subarray}{c}0<y<\psi\left(\bar{x}\right)\\\
\left|\alpha\right|\leq m\\\
i=1,\cdots,i_{\max}\end{subarray}}\left|y\right|^{\left|\alpha\right|-m}\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{j}-Q_{i}\right\\}\left(\bar{x},y\right)\right|$
$\displaystyle\leq$ $\displaystyle
C\max_{\begin{subarray}{c}\sigma=1,\cdots,\sigma_{\max}\\\
\left|\alpha\right|\leq m\\\
i=1,\cdots,i_{\max}\end{subarray}}\left|y_{\sigma}\left(\bar{x}\right)\right|^{\left|\alpha\right|-m}\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}P_{j}+B_{j}-Q_{i}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right|\text{.}$

with $C$ depending only on $i_{\max},j_{\max},m$.

Fix $\bar{x}\in(0,\delta)$, and let
$(\vec{p},\vec{q})=\left(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}}\right)\in\mathfrak{p}^{j_{\max}+i_{\max}}$.
Thus, each $p_{j}$ and $q_{i}$ is a polynomial in $y$ of degree at most $m$.
For $0\leq a\leq m,$ $\sigma=1,\cdots,\sigma_{\max},$ $i=1,\cdots,i_{\max},$
let

$\displaystyle\mu_{a,\sigma,i}^{\\#}\left[\bar{x}\right]\left(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}}\right)$
$\displaystyle=$
$\displaystyle\left.\left(y_{\sigma}\left(\bar{x}\right)\right)^{a-m}\partial_{y}^{a}\left\\{\sum_{j}A_{ij}\left(\bar{x},y\right)p_{j}\left(y\right)+B_{i}\left(\bar{x},y\right)-q_{i}\left(y\right)\right\\}\right|_{y=y_{\sigma}\left(\bar{x}\right)}.$

Note that we don’t take $x$-derivatives here, only $y$-derivatives.

The $\mu_{a,\sigma,i}^{\\#}(\bar{x})$ are affine functions from
$\mathfrak{p}^{j_{\max}+i_{\max}}$ to $\mathbb{R}$; thus, each
$\mu_{a,\sigma,i}^{\\#}(\bar{x})$ belongs to Aff. Let
$\mu_{1}\left(\bar{x}\right),\cdots,\mu_{l_{\max}}\left(\bar{x}\right)$ be an
enumeration of the $\mu_{a,\sigma,i}^{\\#}\left(\bar{x}\right)$, together with
the linear maps

$\displaystyle\left(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}}\right)$
$\displaystyle\mapsto$
$\displaystyle\left(\bar{x}\right)^{a-m}\partial_{y}^{a}p_{j}\left(0\right)$
$\displaystyle\left(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}}\right)$
$\displaystyle\mapsto$
$\displaystyle\left(\bar{x}\right)^{a-m}\partial_{y}^{a}q_{i}\left(0\right)\text{.}$

We will prove the following

  * (69)

Let $\vec{F},\vec{G}$ be as assumed in ((53)). Then, as $\bar{x}$ varies over
$\left(0,\delta\right)$, the
$\left[\mu_{l}\left(\bar{x}\right)\right]\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\bar{G}\right)$
remain bounded, and these quantities tend to zero as $\bar{x}$ tends to
$0^{+}$.

To prove ((69)), we recall that

$\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}=0\text{,}$

hence

$\displaystyle\mu_{a,\sigma,i}^{\\#}\left(\bar{x}\right)\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G}\right)$
$\displaystyle=$
$\displaystyle-\left(y_{\sigma}\left(\bar{x}\right)\right)^{a-m}\left.\partial_{y}^{a}\left[\sum_{i}A_{ij}\left(\bar{x},y\right)\left\\{F_{j}\left(\bar{x},y\right)-j_{\bar{x}}F_{j}\left(y\right)\right\\}-\left\\{G_{j}\left(\bar{x},y\right)-j_{\bar{x}}G_{j}\left(y\right)\right\\}\right]\right|_{y=y_{\sigma}}.$

Let
$w_{F}\left(\bar{x}\right)=\max_{\left|\beta\right|=m,j=1,\cdots,j_{\max}}\left(\sup_{0\leq
y\leq\psi\left(\bar{x}\right)}\left[\partial^{\beta}F_{j}\left(\bar{x},y\right)\right]-\inf_{0\leq
y\leq\psi\left(\bar{x}\right)}\left[\partial^{\beta}F_{j}\left(\bar{x},y\right)\right]\right)$
and similarly define $w_{G}\left(\bar{x}\right)$ as above, with $G$ in place
of $F.$

Because $\vec{F},\vec{G}$ belong to
$C^{m}\left({\Omega}^{\text{closure}}_{\delta},\mathbb{R}^{j_{\max}}\right)$
and
$C^{m}\left({\Omega}^{\text{closure}}_{\delta},\mathbb{R}^{i_{\max}}\right)$
respectively, while $\psi\left(\bar{x}\right)\rightarrow 0$ as
$\bar{x}\rightarrow 0,$ we know that $w_{F}\left(\bar{x}\right)$,
$w_{G}\left(\bar{x}\right)$ are bounded as $\bar{x}$ varies over
$\left(0,\delta\right)$, and moreover
$w_{F}\left(\bar{x}\right),w_{G}\left(\bar{x}\right)\rightarrow 0$ as
$\bar{x}\rightarrow 0^{+}$.

Taylor’s theorem gives

  * (72)

$\left|\partial_{y}^{a}\left[F_{j}\left(\bar{x},y\right)-j_{\bar{x}}F_{j}\left(y\right)\right]\right|\leq
Cw_{F}\left(\bar{x}\right)\cdot y^{m-a}$ for $0\leq a\leq m$,
$0<y<\psi\left(\bar{x}\right),$ $j=1,\cdots,j_{\max}$.

  * (74)

$\left|\partial_{y}^{a}\left\\{G_{i}\left(\bar{x},y\right)-j_{\bar{x}}G_{i}\left(y\right)\right\\}\right|\leq
Cw_{G}\left(\bar{x}\right)\cdot y^{m-a}$ for $0\leq a\leq
m,0<y<\psi\left(\bar{x}\right),i=1,\cdots,i_{\max}$.

We recall that

  * (76)

$|\partial_{y}^{a}A_{ij}(\bar{x},y)|\leq Cy^{-a}$ for $0\leq a\leq
m,0<y<\psi\left(\bar{x}\right),i=1,\cdots,i_{\max},j=1,\cdots,j_{\max}$.

Putting ((72)),((74)),((76)) into (6.1), we find that

$\left|\mu_{a,\sigma,i}^{\\#}\left(\bar{x}\right)\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G}\right)\right|\leq
Cw_{F}\left(\bar{x}\right)+Cw_{G}\left(\bar{x}\right)\text{,}$

hence the
$\mu_{a,\sigma,i}^{\\#}\left(\bar{x}\right)\left(j_{\bar{x}}\vec{F},J_{\bar{x}}\vec{G}\right)$
remain bounded as $\bar{x}$ varies over $\left(0,\delta\right)$, and these
quantities tend to zero as $\bar{x}\rightarrow 0^{+}$.

Also, because $J_{\left(0,0\right)}\vec{F}=0,J_{\left(0,0\right)}\vec{G}=0,$
and $\vec{F},\vec{G}$ are in
$C^{m}\left({\Omega}_{\delta}^{\text{closure}},\mathbb{R}^{j_{\max}}\right)$
and
$C^{m}\left({\Omega}_{\delta}^{\text{closure}},\mathbb{R}^{i_{\max}}\right)$
respectively, we see that

$\left(\bar{x}\right)^{a-m}\partial_{y}^{a}F_{j}\left(\bar{x},0\right),\left(\bar{x}\right)^{a-m}\partial_{y}^{a}G_{i}\left(\bar{x},0\right),$

for $0\leq a\leq m$, remain bounded as $\bar{x}$ varies over
$\left(0,\delta\right),$ and these quantities tend to zero as
$\bar{x}\rightarrow 0^{+}$.

Thus, all the
$\mu_{l}\left(\bar{x}\right)\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G}\right)$
remain bounded on $\left(0,\delta\right)$ and tend to zero as
$\bar{x}\rightarrow 0^{+}$.

We have proven ((69)). Thus, we have defined our
$\lambda_{1},\cdots,\lambda_{k_{\max}}$ and $\mu_{1},\cdots,\mu_{l_{\max}}$
and we have proven ((53)).

We now set out to prove ((55)).

Thus, let $\vec{F}^{\\#}=\left(F_{1}^{\\#},\cdots,F_{j_{\max}}^{\\#}\right)$
and $\vec{G}^{\\#}=\left(G_{1}^{\\#},\cdots,G_{i_{\max}}^{\\#}\right)$ be as
in ((55)).

Recall, each $F_{j}^{\\#}$ and $G_{i}^{\\#}$ is a semialgebraic map from
$(0,\delta)$ into $\mathfrak{p}$, and moreover

$\left[\lambda_{k}\left(\bar{x}\right)\right]\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)=0\text{
for }k=1,\cdots,k_{\max},\text{ all }\bar{x}\in\left(0,\delta\right);\text{
and}$

$\left[\mu_{l}\left(\bar{x}\right)\right]\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)$
is bounded as $\bar{x}$ varies over $\left(0,\delta\right)$ and tends to zero
as $\bar{x}\rightarrow 0^{+}$ for each $l=1,\cdots,l_{\max}$.

Then

  * (78)

$F_{j}^{\\#}\left(\bar{x}\right)$ has the form
$y\mapsto\sum_{s=0}^{m}F_{js}\left(\bar{x}\right)y^{s}$ and

  * (80)

$G_{i}^{\\#}\left(\bar{x}\right)$ has the form
$y\mapsto\sum_{s=0}^{m}G_{is}\left(\bar{x}\right)y^{s}$,

with $F_{js},G_{is}$ semialgebraic functions of one variable. Taking $\delta$
small (depending on $\vec{F}^{\\#},\vec{G}^{\\#}$), we may assume the
$F_{js},G_{is}$ are $C^{\infty}$ on $(0,\delta)$.

Now, we define $\vec{F}=\left(F_{1},\cdots,F_{j_{\max}}\right)$,
$\vec{G}=\left(G_{1},\cdots,G_{i_{\max}}\right),\vec{G}^{\\#\\#}=\left(G_{1}^{\\#\\#},\cdots,G_{i_{\max}}^{\\#\\#}\right),$
where

(82)
$F_{j}\left(\bar{x},y\right)=\sum_{s=0}^{m}F_{js}\left(\bar{x}\right)y^{s}$

for $\left(\bar{x},y\right)\in\left(0,\delta\right)\times\mathbb{R}$,
$j=1,\cdots,j_{\max},$

(83)
$G_{i}^{\\#\\#}\left(\bar{x},y\right)=\sum_{s=0}^{m}G_{is}\left(\bar{x}\right)y^{s}$

for $\left(\bar{x},y\right)\in\left(0,\delta\right)\times\mathbb{R}$,
$i=1,\cdots,i_{\max}$,

$G_{i}\left(\bar{x},y\right)=\sum_{j}A_{ij}\left(\bar{x},y\right)F_{j}\left(\bar{x},y\right)+B_{i}\left(\bar{x},y\right)$

for $\left(\bar{x},y\right)\in\Omega_{\delta}$, $i=1,\cdots,i_{\max}$.

Note that $F_{j},G_{i}^{\\#\\#}$ are $C^{\infty}$ functions on
$\left(0,\delta\right)\times\mathbb{R}$ because the $F_{js},G_{is}$ are
$C^{\infty}$ functions on $\left(0,\delta\right)$.

The functions $F_{j},G_{i}^{\\#\\#},G_{i}$ are semialgebraic because
$F_{j}^{\\#},G_{i}^{\\#}$ are semialgebraic.

Let $\bar{x}\in\left(0,\delta\right)$. Then

(84)
$j_{\bar{x}}F_{j}=F_{j}^{\\#}\left(\bar{x}\right)\in\mathfrak{p},j_{\bar{x}}G_{i}^{\\#\\#}=G_{i}^{\\#}\left(\bar{x}\right)\in\mathfrak{p}.$

Therefore, for all $\bar{x}$ in a small neighborhood of a given
$\overline{\overline{x}}\in\left(0,\delta\right)$, we have

$\lambda_{k}(\bar{x})\left(j_{\bar{x}}\vec{F},j_{\bar{x}}\vec{G}^{\\#\\#}\right)=\lambda_{k}(\bar{x})\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)=0$

for $k=1,\cdots,k_{\max}$; the last equality is an assumption made in ((55)).

Because $\vec{F},\vec{G}^{\\#\\#}$ are $C^{\infty}$ in a neighborhood of
$\left(\overline{\overline{x}},0\right),$ the defining property of the
$\lambda_{k}$ now tells us that

$\left(J_{\left(\bar{x},0\right)}\vec{F},J_{\left(\bar{x},0\right)}\vec{G}^{\\#\\#}\right)\in
H\left(\bar{x},0\right)$

for all $\bar{x}$ in a small neighborhood of $\overline{\overline{x}}$.

Recalling that $\overline{\overline{x}}\in\left(0,\delta\right)$ is arbitrary,
we conclude that

(85)
$\left(J_{\left(\bar{x},0\right)}\vec{F},J_{\left(\bar{x},0\right)}\vec{G}^{\\#\\#}\right)\in
H\left(\bar{x},0\right)\text{ for all }\bar{x}\in\left(0,\delta\right).$

By definition of $H\left(\bar{x},0\right)$ and by the estimates

$\displaystyle\partial^{\alpha}\left(F_{j}-J_{\left(\bar{x},0\right)}F_{j}\right)\left(\bar{x},y\right)$
$\displaystyle=$ $\displaystyle o\left(y^{m-\left|\alpha\right|}\right),$
$\displaystyle\partial^{\alpha}\left(G_{i}^{\\#\\#}-J_{\left(\bar{x},0\right)}G_{i}^{\\#\\#}\right)\left(\bar{x},y\right)$
$\displaystyle=$ $\displaystyle o\left(y^{m-\left|\alpha\right|}\right),\text{
and}$ $\displaystyle\left|\partial^{\alpha}A_{ij}\left(x,y\right)\right|$
$\displaystyle\leq$ $\displaystyle Cy^{-\left|\alpha\right|}\text{,}$

we therefore have the following:

  * (86)

For any $\bar{x}\in\left(0,\delta\right),$ any $i=1,\cdots,i_{\max},$ and any
$\left|\alpha\right|\leq m$, the quantity

$y^{\left|\alpha\right|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y\right)$

is bounded as $y$ varies over $\left(0,\psi\left(\bar{x}\right)\right)$ and
tends to zero as $y\rightarrow 0^{+}$.

We don’t yet know that the above convergence is uniform in $\bar{x}.$

Next, we recall from ((55)) the assumption that the
$\mu_{l}\left(\bar{x}\right)\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)$
remain bounded as $\bar{x}$ varies over $\left(0,\delta\right)$ and moreover
these quantities tend to zero as $\bar{x}\rightarrow 0^{+}$.

Thus, the quantities

(88)
$\left(y_{\sigma}\left(\bar{x}\right)\right)^{a-m}\partial_{y}^{a}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)$

for $0\leq a\leq m,i=1,\cdots,i_{\max},\sigma=1,\cdots,\sigma_{\max}$, remain
bounded as $\bar{x}$ varies over $\left(0,\delta\right)$, and tend to zero as
$\bar{x}\rightarrow 0^{+}$.

Because those quantities are semialgebraic functions of one variable, we may
pass to a smaller $\delta$ and assert for any $b$, say $0\leq b\leq m$, that

(89)
$\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{y_{\sigma}\left(\bar{x}\right)^{a-m}\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}=o\left(\bar{x}^{-b}\right)$

as $\bar{x}\rightarrow 0^{+}$ and this quantity is bounded for $\bar{x}$
bounded away from $0$.

For $0\leq a+b\leq m,$ we will check that

(90)
$\left(\bar{x}\right)^{a+b-m}\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}=o\left(1\right)$

as $\bar{x}\rightarrow 0^{+}$ and the left-hand side is bounded.

To see this, we write

$\displaystyle\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}$
$\displaystyle=$
$\displaystyle\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{\left(y_{\sigma}\left(\bar{x}\right)\right)^{m-a}\left(y_{\sigma}\left(\bar{x}\right)\right)^{a-m}\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}$
$\displaystyle=$
$\displaystyle\sum_{b^{\prime}+b^{\prime\prime}=b}\text{coeff}\left(b^{\prime},b^{\prime\prime}\right)\underset{\left(\dagger\right)}{\underbrace{\left[\left(\frac{d}{d\bar{x}}\right)^{b^{\prime}}\left(y_{\sigma}\left(\bar{x}\right)\right)^{m-a}\right]}}\cdot$
$\displaystyle\underset{\left(\ddagger\right)}{\underbrace{\left[\left(\frac{d}{d\bar{x}}\right)^{b^{\prime\prime}}\left\\{\left(y_{\sigma}\left(\bar{x}\right)\right)^{a-m}\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}\right]}}\text{.}$

Since $y_{\sigma}\left(\bar{x}\right)$ is given by a Puiseux series for
$\bar{x}\in\left(0,\delta\right)$ (small enough $\delta$),

$\left(\dagger\right)=O\left(y_{\sigma}\left(\bar{x}\right)\right)^{m-a}\cdot\bar{x}^{-b^{\prime}}=O\left(y_{\sigma}\left(\bar{x}\right)^{m-a-b^{\prime}}\right),$

because
$0<y_{\sigma}\left(\bar{x}\right)<\psi\left(\bar{x}\right)\leq\bar{x}$. By
(89), $\left(\ddagger\right)$ is $o\left(\bar{x}^{-b^{\prime\prime}}\right)$
as $\bar{x}\rightarrow 0^{+}$.

So in fact, we get not only (90) but the stronger result

(91)
$\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{\partial_{y}^{a}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}=o\left(y_{\sigma}\left(\bar{x}\right)^{m-a}\cdot\bar{x}^{-b}\right)$

as $\bar{x}\rightarrow 0^{+};$ the left-hand side is bounded.

Introduce the vector field $X_{\sigma}=\frac{\partial}{\partial
x}+y_{\sigma}^{\prime}\left(\bar{x}\right)\frac{\partial}{\partial y}$ on
$\mathbb{R}^{2}.$ We have

$\left(\frac{d}{d\bar{x}}\right)^{b}\left\\{\mathcal{F}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right\\}=\left.\left(X_{\sigma}\right)^{b}\mathcal{F}\right|_{\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)}\text{
for any }\mathcal{F}\in C_{loc}^{b}\left(\mathbb{R}^{2}\right)\text{.}$

Therefore, (91) yields

(92)
$\left(X_{\sigma}^{b}\partial_{y}^{a}\right)\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)=o\left(y_{\sigma}\left(\bar{x}\right)^{m-a}\cdot\bar{x}^{-b}\right)\text{
as }\bar{x}\rightarrow 0^{+}$

and the left-hand side is bounded for all $\bar{x}$, for $a+b\leq
m,\sigma=1,\cdots,\sigma_{\max},i=1,\cdots,i_{\max}$.

This implies that

  * (93)

$\left(y_{\sigma}\left(\bar{x}\right)\right)^{\left|\alpha\right|-m}\partial^{\alpha}\left[\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right]\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)$
is bounded on $(0,\delta)$ and tends to zero as $\bar{x}\rightarrow 0^{+}$,
for $\left|\alpha\right|\leq
m,i=1,\cdots,i_{\max},\sigma=1,\cdots,\sigma_{\max}$.

Let $\alpha=\left(b,a\right),$
$\partial^{\alpha}=\partial_{x}^{b}\partial_{y}^{a}$.

We deduce ((93)) from (92) by induction on $b$. For $b=0,$ ((93)) is the same
as (92).

Assume we know ((93)) for all $b^{\prime}<b.$ We prove ((93)) for the given
$b,$ using our induction hypothesis for $b^{\prime}$, together with (92).

The quantity

(95)
$X_{\sigma}^{b}\partial_{y}^{a}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)$

is a sum of terms of the form

(96)
$\left(\partial_{x}^{b_{1}}y_{\sigma}\left(\bar{x}\right)\right)\cdot\cdots\cdot\left(\partial_{x}^{b_{\nu}}y_{\sigma}\left(\bar{x}\right)\right)\cdot\partial_{x}^{\bar{b}}\partial_{y}^{a+\nu}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)$

with $b_{t}\geq 1$ each $t$, $b_{1}+\cdots+b_{\nu}+\bar{b}=b.$

Note
$\bar{b}+\left(a+\nu\right)=a+\bar{b}+b_{1}+\cdots+b_{\nu}-\left(b_{1}-1\right)-\cdots-\left(b_{\nu}-1\right)\leq
a+b$.

We know that
$\eqref{LHS}=o\left(y_{\sigma}\left(\bar{x}\right)^{m-a-b}\right)$ by (92).

If $\bar{b}<b,$ then by our induction hypothesis, the term (96) is dominated
by

$\displaystyle O\left(\overset{\text{Here again we use
}0<y_{\sigma}<\bar{x}\text{.}}{\overbrace{y_{\sigma}\left(\bar{x}\right)^{-\left[b_{1}-1\right]-\cdots-\left[b_{\nu}-1\right]}}}\right)\cdot
o\left(y_{\sigma}\left(\bar{x}\right)^{m-\left[a+\nu\right]-\bar{b}}\right)$
$\displaystyle=$ $\displaystyle
o\left(y_{\sigma}\left(\bar{x}\right)^{m-a-\bar{b}-b_{1}-\cdots-
b_{\nu}}\right)=o\left(y_{\sigma}\left(\bar{x}\right)^{m-a-b}\right)\text{.}$

Therefore, in the equation $\eqref{LHS}=\sum\eqref{RHS}$, all terms are
$o\left(y_{\sigma}\left(\bar{x}\right)^{m-a-b}\right)$, except possibly the
term arising from $\bar{b}=b$, which is

$\partial_{x}^{b}\partial_{y}^{a}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\text{.}$

Therefore,

$\partial_{x}^{b}\partial_{y}^{a}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)=o\left(y_{\sigma}\left(\bar{x}\right)^{m-a-b}\right)\text{,
as }\bar{x}\rightarrow 0^{+}\text{.}$

This completes our induction on $b$, proving ((93)).

Thus,

  * (97)

$\max_{\begin{subarray}{c}\sigma=1,\cdots,\sigma_{\max}\\\
i=1,\cdots,i_{\max}\\\ \left|\alpha\right|\leq
m\end{subarray}}\left(y_{\sigma}\left(\bar{x}\right)\right)^{\left|\alpha\right|-m}\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)\right|\text{
}$ is bounded on $\left(0,\delta\right)$ and tends to zero as $\bar{x}$ tends
to $0^{+}$.

Recall that our $\mu_{l}(\bar{x})$ include the affine maps
$(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}})\mapsto\bar{x}^{a-m}\partial_{y}^{a}p_{j}(0)$
and
$(p_{1},\cdots,p_{j_{\max}},q_{1},\cdots,q_{i_{\max}})\mapsto\bar{x}^{a-m}\partial_{y}^{a}q_{i}(0)$
for $0\leq a\leq m.$ Our assumption on the $\mu$’s made in ((55)) tells us
therefore that
$\bar{x}^{a-m}\partial_{y}^{a}\left(F_{j}^{\\#}\left(\bar{x}\right)\right)\left(0\right)$
and
$\bar{x}^{a-m}\partial_{y}^{a}\left(G_{i}^{\\#}\left(\bar{x}\right)\right)\left(0\right)$
are bounded on $\left(0,\delta\right)$ and tend to zero as $\bar{x}\rightarrow
0^{+}$.

That is,

  * (99)

$\bar{x}^{s-m}F_{js}\left(\bar{x}\right),\bar{x}^{s-m}G_{is}\left(\bar{x}\right)$
are bounded on $\left(0,\delta\right)$ and tend to zero as $\bar{x}\rightarrow
0^{+}$. $\left(0\leq s\leq m\right)$.

  * (101)

Because $F_{js},G_{js}$ are semialgebraic functions of one variable, it
follows that, for $s,t\leq m$, the functions

$\left(\frac{d}{d\bar{x}}\right)^{t}F_{js}\left(\bar{x}\right),\left(\frac{d}{d\bar{x}}\right)^{t}G_{is}\left(\bar{x}\right)$

are bounded on $\left(0,\delta\right)$ if $s+t\leq m$ and are
$o\left(\bar{x}^{m-s-t}\right)$ as $\bar{x}\rightarrow 0^{+}$ (even if
$s+t>m$).

Recalling now the definitions of the $F_{j}$ and $G_{i}^{\\#\\#}$ in terms of
the $F_{j},G_{is}$ (see (82), (83)), we conclude that

$\displaystyle\partial_{\bar{x}}^{t}\partial_{y}^{s}F_{j}\left(\bar{x},y\right)$
$\displaystyle=$ $\displaystyle\sum_{m\geq\underline{s}\geq
s}\left[\left(\frac{d}{d\bar{x}}\right)^{t}F_{j\underline{s}}\left(\bar{x}\right)\right]\left(\text{coefficient
}\left(\underline{s},s\right)\right)\cdot y^{\underline{s}-s}$
$\displaystyle=$ $\displaystyle\sum_{m\geq\underline{s}\geq
s}o\left(\bar{x}^{m-t-\underline{s}}\right)\cdot y^{\underline{s}-s}\text{.}$

If $s+t=m,$ then this is equal to
$o\left(\frac{y}{\bar{x}}\right)^{\underline{s}-s}=o\left(1\right)$ for
$0<y<\psi\left(\bar{x}\right)\leq\bar{x}$.

Therefore, for $\left|\beta\right|=m,$ we have
$\left|\partial^{\beta}F_{j}\left(\bar{x},y\right)\right|=o\left(1\right)$ as
$\left(\bar{x},y\right)\in\Omega_{\delta}$ tends to zero.

Similarly,
$\left|\partial^{\beta}G_{i}^{\\#\\#}\left(\bar{x},y\right)\right|=o\left(1\right)$
as $\left(\bar{x},y\right)\in\Omega_{\delta}$ tends to zero.

That is, for $\left|\beta\right|=m$, the functions
$\partial^{\beta}F_{j}\left(\bar{x},y\right)$ and
$\partial^{\beta}G_{i}^{\\#\\#}\left(\bar{x},y\right)$ are bounded on
$\Omega_{\delta}$ and they tend to zero as $\bar{x}\rightarrow 0^{+}$ (keeping
$\left(\bar{x},y\right)\in\Omega_{\delta}$).

Let
$\mathcal{E}\left(\bar{x}\right)=\sup\left\\{\left|\partial^{\beta}F_{j}\left(\bar{x},y\right)\right|,\left|\partial^{\beta}G_{i}^{\\#\\#}\left(\bar{x},y\right)\right|:\left|\beta\right|=m,0<y<\psi\left(\bar{x}\right)\text{
(all }i,j)\right\\}.$

Then

(103) $\mathcal{E}\left(\bar{x}\right)\text{is bounded on
}\left(0,\delta\right)\text{ and tends to zero as }\bar{x}\rightarrow 0^{+}.$

By Taylor’s theorem,

$\left|\partial^{\alpha}\left\\{F_{j}-J_{\left(\bar{x},0\right)}F_{j}\right\\}\left(\bar{x},y\right)\right|\leq
Cy^{m-\left|\alpha\right|}\mathcal{E}\left(\bar{x}\right)\text{ for
}\left|\alpha\right|\leq m,\left(\bar{x},y\right)\in\Omega_{\delta}\text{.}$

Recall that

$\left|\partial^{\alpha}A_{ij}\left(\bar{x},y\right)\right|\leq
Cy^{-\left|\alpha\right|}\text{ for }\left|\alpha\right|\leq m\text{ and
}\left(\bar{x},y\right)\in\Omega_{\delta}\text{.}$

Just as we estimated the functions $F_{j}$ above, we have from Taylor’s
theorem that

$\left|\partial^{\alpha}\left\\{G_{i}^{\\#\\#}-J_{\left(\bar{x},0\right)}G_{i}^{\\#\\#}\right\\}\left(\bar{x},y\right)\right|\leq
Cy^{m-\left|\alpha\right|}\mathcal{E}\left(\bar{x}\right)\text{ for
}\left|\alpha\right|\leq m,\left(\bar{x},y\right)\in\Omega_{\delta}\text{.}$

Combining these estimates, we see that

(104)
$\displaystyle\left|\partial^{\alpha}\left\\{\sum_{j}A_{ij}\left(F_{j}-J_{\left(\bar{x},0\right)}F_{j}\right)-\left(G_{i}^{\\#\\#}-J_{\left(\bar{x},0\right)}G_{i}^{\\#\\#}\right)\right\\}\left(x,y\right)\right|$
$\displaystyle\leq$ $\displaystyle
Cy^{m-\left|\alpha\right|}\mathcal{E}\left(\bar{x}\right)\text{ for
}\left|\alpha\right|\leq m,\left(\bar{x},y\right)\in\Omega_{\delta}\text{.}$

Combining ((97)), (103), (104), we see that

(105)
$\displaystyle\left(y_{\sigma}\left(\bar{x}\right)\right)^{\left|\alpha\right|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}\left[J_{\left(\bar{x},0\right)}F_{j}\right]+B_{i}-\left[J_{\left(\bar{x},0\right)}G_{i}^{\\#\\#}\right]\right\\}\left(\bar{x},y_{\sigma}\left(\bar{x}\right)\right)$
$\displaystyle\text{is bounded on }\left(0,\delta\right)\text{ and tends to 0
as }\bar{x}\text{ tends to }0^{+}\text{.}$

Recall that
$\left(J_{(\bar{x},0)}\vec{F},J_{\left(\bar{x},0\right)}\vec{G}^{\\#\\#}\right)\in
H\left(\bar{x}\right)$ for all $\bar{x}\in(0,\delta]$ (see (85)).

The above results, together with the property ((67)) of the
$y_{\sigma}\left(\bar{x}\right)$ now tells us that

  * (106)

$y^{\left|\alpha\right|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}\left(J_{\left(\bar{x},0\right)}F_{j}\right)+B_{i}-\left(J_{\left(\bar{x},0\right)}G_{i}^{\\#\\#}\right)\right\\}\left(\bar{x},y\right)$
is bounded on $\Omega_{\delta}$ and tends to zero as
$\left(\bar{x},y\right)\in\Omega_{\delta}$ tends to zero.

Together with (103), (104), this yields the following result

  * (108)

$y^{\left|\alpha\right|-m}\partial^{\alpha}\left\\{\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right\\}\left(\bar{x},y\right)$
is bounded on $\Omega_{\delta}$ and tends to zero as
$\left(\bar{x},y\right)\in\Omega_{\delta}$ tends to zero. Here,
$i=1,\cdots,i_{\max}$ and $|\alpha|\leq m$ are arbitrary.

From ((86)), we have

  * (110)

$\lim_{y\rightarrow
0^{+}}y^{|\alpha|-m}\partial^{\alpha}\left(\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right)(x,y)=0$
for each fixed $x\in(0,\delta)$.

The functions $A_{ij},F_{j},B_{i},G_{i}^{\\#\\#}$ are semialgebraic.
Therefore, by Lemma 3.3, there exist a positive integer $K$ and a
semialgebraic function of one variable $\mathcal{A}(x)$ such that

  * (112)

$\left|y^{|\alpha|-m}\partial^{\alpha}\left(\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right)(x,y)\right|\leq\mathcal{A}(x)\cdot
y^{\frac{1}{K}}$ for all $(x,y)\in\Omega_{\delta}$, $|\alpha|\leq
m,i=1,\cdots,i_{\max}$.

Taking $\delta$ smaller, we may assume $\mathcal{A}(x)$ is $C^{\infty}$ on
$(0,\delta]$.

Consequently,
$y^{|\alpha|-m}\partial^{\alpha}\left(\sum_{j}A_{ij}F_{j}+B_{i}-G_{i}^{\\#\\#}\right)(x,y)$
tends to zero as $y\rightarrow 0^{+}$, uniformly as $x$ varies over
$(\varepsilon,\delta)$ for any $\varepsilon>0$.

Recalling that $G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$, we see that for
$\left|\alpha\right|\leq m,i=1,\cdots,i_{\max},$

(114)
$y^{|\alpha|-m}\partial^{\alpha}\left\\{G_{i}-G_{i}^{\\#\\#}\right\\}\left(x,y\right)\rightarrow
0$

as $y\rightarrow 0^{+}$ uniformly for $x$ in each interval
$\left(\varepsilon,\delta\right)$.

Recalling that $G_{i}^{\\#\\#}$ belongs to $C^{\infty}$ in a neighborhood of
$\left(x,0\right)$ (each $x\in\left(0,\delta\right)$), we conclude that the
derivatives $\partial^{\alpha}G_{i}\left(x,y\right)$ ($\left|\alpha\right|\leq
m,i=1,\cdots,i_{\max}$), initially defined on
$\Omega_{\delta}=\left\\{\left(x,y\right):0<x<\delta,0<y<\psi\left(x\right)\right\\}$
extend to continuous functions on

(115) $\Omega_{\delta}^{++}\equiv\left\\{\left(x,y\right):0<x<\delta,0\leq
y<\psi\left(x\right)\right\\}.$

Next, recall that $F_{js}$ is $C^{\infty}$ on $\left(0,\delta\right)$ and that
we assume that
$\left|\partial^{\alpha}A_{ij}\left(x,y\right)\right|,\left|\partial^{\alpha}B_{i}\left(x,y\right)\right|\leq
Cy^{-\left|\alpha\right|}$ on

(116)
$\Omega^{+}=\left\\{\left(x,y\right):0<x<\delta,0<y\leq\psi\left(x\right)\right\\}$

on which the functions $\partial^{\alpha}A_{ij},\partial^{\alpha}B_{i}$ are
assumed to be continuous.

We defined

$\displaystyle G_{i}$ $\displaystyle=$
$\displaystyle\sum_{j}A_{ij}F_{j}+B_{i}$ $\displaystyle=$
$\displaystyle\sum_{j}A_{ij}\left(x,y\right)\left[\sum_{s=0}^{m}F_{js}\left(x\right)y^{s}\right]+B_{i}\left(x,y\right)\text{.}$

The above remarks (and the fact that $\psi\left(x\right)\not=0$ for
$x\in\left(0,\delta\right)$) show that $\partial^{\alpha}G_{i}$ extends to a
continuous function on $\Omega^{+}$ (see (116)), for $\left|\alpha\right|\leq
m,i=1,\cdots,i_{\max}$.

Combining our results for $\Omega^{+}$ (see (116)) and for $\Omega^{++}$ (see
(115)), we see that $\partial^{\alpha}G_{i}$ extends to a continuous function
on
$\Omega_{\frac{2\delta}{3}}^{\text{closure}}\setminus\left\\{\left(0,0\right)\right\\}$
for each $i=1,\cdots,i_{\max},\left|\alpha\right|\leq m$.

Also, $\partial^{\alpha}F_{i}$ is a continuous function on
$\Omega_{\frac{2}{3}\delta}^{\text{closure}}\setminus\left\\{\left(0,0\right)\right\\}$
because $F_{i}$ is $C^{\infty}$ on $\left(0,\delta\right)\times\mathbb{R}$.

By ((99)), we have $G_{is}(x)=o(x^{m-s})$ ($0\leq s\leq m$) on $(0,\delta)$.
Because $G_{is}$ is semialgebraic, it follows that after possibly reducing
$\delta$, we have

$\left(\frac{d}{dx}\right)^{t}G_{is}\left(x\right)=o\left(x^{m-s-t}\right)\text{
for }0\leq t\leq m,0\leq s\leq m,i=1,\cdots,i_{\max}\text{.}$

Because
$G_{i}^{\\#\\#}\left(x,y\right)=\sum_{\underline{s}=0}^{m}G_{i\underline{s}}\left(x\right)y^{\underline{s}}$
and $0<y<\psi\left(x\right)\leq x$ on $\Omega_{\delta}$, we have on
$\Omega_{\delta}$ that

$\displaystyle\left|\partial_{x}^{t}\partial_{y}^{s}G_{i}^{\\#\\#}\left(x,y\right)\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{\underline{s}=s}^{m}\text{coeff}\left(\underline{s},s\right)\cdot\left(\frac{d}{dx}\right)^{t}G_{i\underline{s}}\left(x\right)\cdot
y^{\underline{s}-s}\right|$ $\displaystyle=$ $\displaystyle
o\left(\sum_{\underline{s}=s}^{m}x^{m-\underline{s}-t}\cdot
y^{\underline{s}-s}\right)$ $\displaystyle=$ $\displaystyle
o\left(\sum_{\underline{s}=s}^{m}x^{m-\underline{s}-t}\cdot
x^{\underline{s}-s}\right)$ $\displaystyle=$ $\displaystyle
o\left(x^{m-s-t}\right)\text{ on }\Omega_{\delta}\text{ for }s,t\leq
m\text{.}$

In particular,

  * (117)

$\partial^{\alpha}G_{i}^{\\#\\#}\left(x,y\right)\rightarrow 0$ as
$\left(x,y\right)\in\Omega_{\delta}$ tends to $\left(0,0\right)$ for
$\left|\alpha\right|\leq m,i=1,\cdots,i_{\max}$.

On the other hand, recalling the definition $G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$,
we see from ((108)) that
$\partial^{\alpha}\left(G_{i}-G_{i}^{\\#\\#}\right)\left(x,y\right)\rightarrow
0$ as $\left(x,y\right)\in\Omega_{\delta}$ tends to $\left(0,0\right)$ for
each $\left|\alpha\right|\leq m$. Together with ((117)), this shows that
$\partial^{\alpha}G_{i}\left(x,y\right)\rightarrow 0$ as
$\left(x,y\right)\in\Omega_{\delta}$ tends to $\left(0,0\right)$ for each
$\left|\alpha\right|\leq m$.

Next, recall from ((99)) that $F_{js}(x)=o(x^{m-s})$ for $x\in(0,\delta)$,
$j=1,\cdots,j_{\max},s=0,\cdots,m$.

Because the $F_{jk}$ are semialgebraic functions of one variable, we conclude
(after reducing $\delta$) that
$\left(\frac{d}{dx}\right)^{t}F_{js}\left(x\right)=o\left(x^{m-s-t}\right)$ on
$\left(0,\delta\right)$ for $t\leq m$.

Now, for $s+t\leq m$ and $\left(x,y\right)\in\Omega_{\delta}$ (hence
$0<y<\psi\left(x\right)\leq x$), we have

$\displaystyle\left|\left(\frac{\partial}{\partial
y}\right)^{s}\left(\frac{\partial}{\partial
x}\right)^{t}F_{j}\left(x,y\right)\right|$ $\displaystyle=$
$\displaystyle\left|\left(\frac{\partial}{\partial
y}\right)^{s}\left(\frac{\partial}{\partial
x}\right)^{t}\sum_{\underline{s}=0}^{m}F_{j\underline{s}}\left(x\right)y^{\underline{s}}\right|$
$\displaystyle=$
$\displaystyle\left|\sum_{\underline{s}=s}^{m}\text{coeff}\left(\underline{s},s\right)\left[\left(\frac{d}{dx}\right)^{t}F_{j\underline{s}}\left(x\right)\right]\cdot
y^{\underline{s}-s}\right|$ $\displaystyle\leq$ $\displaystyle
C\sum_{\underline{s}=s}^{m}\left|\left(\frac{d}{dx}\right)^{t}F_{j\underline{s}}\left(x\right)\right|\cdot
x^{\underline{s}-s}$ $\displaystyle=$ $\displaystyle
o\left(\sum_{\underline{s}=0}^{m}x^{m-\underline{s}-t}x^{\underline{s}-s}\right)=o\left(x^{m-s-t}\right)\text{.}$

Thus, for $\left|\alpha\right|\leq m$, and $j=1,\cdots,j_{\max},$ we have

$\partial^{\alpha}F_{j}\left(x,y\right)\rightarrow 0\text{ as
}\left(x,y\right)\in\Omega_{\delta}\text{ tends to }\left(0,0\right)\text{.}$

We now know the following: $G_{i}=\sum_{j}A_{ij}F_{j}+B_{i}$ on
$\Omega_{\delta}.$ The $F_{j}$ and $G_{i}$ are semialgebraic on
$\Omega_{\delta}$

For $\left|\alpha\right|\leq m$, the derivatives
$\partial^{\alpha}F_{j},\partial^{\alpha}G_{i}$ extend to continuous functions
on
$\Omega_{2\delta/3}^{\text{closure}}\setminus\left\\{\left(0,0\right)\right\\}$.
For $\left|\alpha\right|\leq m,$ the derivatives
$\partial^{\alpha}F_{j}\left(z\right)$, $\partial^{\alpha}G_{i}\left(z\right)$
tend to zero as $z\in\Omega_{\delta}$ tends to zero.

It follows that the $F_{j}$ and $G_{i}$ extend from $\Omega_{\delta/2}$ to
semialgebraic functions in
$C^{m}\left(\Omega_{\delta/2}^{\text{closure}}\right)$ and those functions all
have $m$-jet zero at the origin. We extend $F_{j},G_{i}$ to semialgebraic
$C^{m}_{loc}$ functions on $\mathbb{R}^{2}$, using Corollary 3.2.

Next, we show that
$j_{\bar{x}}\left(\vec{F},\vec{G}\right)=\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)$
for $\bar{x}\in\left(0,\delta\right)$.

From (84), we have

$j_{\bar{x}}\left(\vec{F},\vec{G}^{\\#\\#}\right)=\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)\text{.}$

From (114), we see that $j_{\bar{x}}\left(G_{i}-G_{i}^{\\#\\#}\right)=0$ for
all $\bar{x}\in\left(0,\delta\right)$. Therefore,

$j_{\bar{x}}\left(\vec{F},\vec{G}\right)=j_{\bar{x}}\left(\vec{F},\vec{G}^{\\#\\#}\right)=\left(\vec{F}^{\\#}\left(\bar{x}\right),\vec{G}^{\\#}\left(\bar{x}\right)\right)\text{,}$

as desired.

Thus, we have proven ((55)).

The proof of Lemma 6.2 is complete.

###  6.2 Patching near a cusp

###### Lemma 6.4

Let $\psi(x)$ be a semialgebraic function on $[0,\delta]$, satisfying
$\psi(0)=0,0<\psi(x)\leq x$ for all $x\in(0,\delta]$. We set

$E_{\delta}=\\{(x,y)\in\mathbb{R}^{2}:0\leq x\leq\delta,0\leq
y\leq\psi(x)\\},$ $E_{\delta}^{+}=\\{(x,y)\in\mathbb{R}^{2}:0\leq
x\leq\delta,\frac{1}{3}\psi(x)\leq y\leq\psi(x)\\},\text{ and }$
$E_{\delta}^{-}=\\{(x,y)\in\mathbb{R}^{2}:0\leq x\leq\delta,0\leq
y\leq\frac{2}{3}\psi(x)\\}.$

Fix a semialgebraic function of one variable, $\theta\left(t\right)$,
satisfying $0\leq\theta\left(t\right)\leq 1$, $\theta\left(t\right)=1$ for
$t\leq 1/3,$ $\theta\left(t\right)=0$ for $t\geq 2/3$, $\theta\in C^{m+100}$.

Then set

$\theta_{-}\left(x,y\right)=\theta\left(\frac{y}{\psi\left(x\right)}\right)\text{,
}\theta_{+}\left(x,y\right)=1-\theta_{-}\left(x,y\right)\text{ for
}\left(x,y\right)\in E_{\delta}\setminus\\{(0,0)\\}\text{.}$

Thus, $\theta_{+},\theta_{-}\geq 0$ and $\theta_{+}+\theta_{-}=1$ on
$E_{\delta}\setminus\\{(0,0)\\}$.

Let $F^{+}\in C^{m}(E_{\delta}^{+})$ and $F^{-}\in C^{m}(E_{\delta}^{-})$ be
semialgebraic functions, with $J_{(0,0)}F^{+}=J_{(0,0)}F^{-}=0$.

Suppose that

(119)
$\partial_{y}^{l}F^{+}(x,\psi(x))-\sum_{j=0}^{m-l}\frac{1}{j!}\partial_{y}^{l+j}F^{-}(x,0)\cdot(\psi(x))^{j}=o((\psi(x))^{m-l})$

as $x\rightarrow 0^{+}$ for each $l=0,\cdots,m$.

Define $F=\theta_{+}\cdot F^{+}+\theta_{-}\cdot F^{-}$ on
$E_{\delta}\setminus\\{(0,0)\\},F(0,0)=0$.

Then $F$ is a $C^{m}$ semialgebraic function on $E_{\delta^{\prime}}$ for some
small $\delta^{\prime}$. The jet of $F$ at the origin is zero. Moreover,
$F=F^{+}$ in a neighborhood of any point $(x,\psi(x))$, $0<x<\delta^{\prime}$;
and $F=F^{-}$ in a neighborhood of any point $(x,0),0<x<\delta^{\prime}$.

Proof. Because $0\leq\psi(x)\leq x$ and $\psi$ is given near $0$ by a
convergent Puiseux series, we have $\psi^{(k)}(x)=O(x^{1-k})$ as $x\rightarrow
0^{+}$, for $k=0,\cdots,m+100$. Also, because $F^{+},F^{-}$ have zero jet at
$(0,0)$, we have, for $|\alpha|=m$, $\partial^{\alpha}F^{+}(x,y)=o(1)$ as
$(x,y)\in E_{\delta}^{+}$ tends to zero and $\partial^{\alpha}F^{-}(x,y)=o(1)$
as $(x,y)\in E_{\delta}^{-}$ tends to zero.

By induction on $\mu$, we now prove that