i\end{subarray}}|a_{jk}(u)|\leq\frac{{\mathtt{M}}(r,n,m,k)}{\lambda^{k-1}}\ .$
(8.2)

By Theorem 4.1, the constant ${\mathtt{M}}={\mathtt{M}}(r,n,m,k)$ in (8.2) can
be explicitly computed and it is uniform for all curves $\gamma\in\Theta_{m}$.

As we have shown in the proof of Lemma 6.5, for any given
$P\in\mathcal{P}(r,m)$, the threshold $\lambda_{0}$ \- if it exists - goes to
zero with the distance of $P$ to the "bad" set $\sigma(r,s,m)$. Therefore, by
formula (8.2), the Taylor coefficients of the curves $\gamma$ on which the
$s$-vanishing condition must be tested may take any value, except for those of
order one which, independently from the choice of $\lambda>0$, are always
uniformly bounded by ${\mathtt{M}}(r,n,m,1)$.

Inspired by the above reasonings, with the notations in (8.2), we give the
following

######  Definition 8.1.

For $i=1,\dots,m$, we introduce the sets

$\displaystyle\begin{split}&\widehat{\Theta}_{m}^{i}:=\left\\{\gamma\in\Theta_{m}^{i}|\max_{\begin{subarray}{c}j\in\\{1,\dots,m\\}\\\
j\neq i\end{subarray}}\\{|a_{j1}(0)|\\}\leq{\mathtt{M}}(r,n,m,1)\right\\}\\\
\end{split}$ (8.3)

and

$\widehat{\Theta}_{m}:=\bigcup_{i=1}^{m}\widehat{\Theta}_{m}^{i}\ ,$ (8.4)

and we denote respectively by
$\widehat{\vartheta}^{i}_{m}(s)\subset\vartheta^{i}_{m}(s)$ and by
$\widehat{\vartheta}_{m}(s)\subset\vartheta_{m}(s)$ their associated subsets
of $s$-truncations.

######  Remark 8.1.

By formula (8.3), if one introduces the decomposition

$\widehat{\vartheta}^{i}_{m}(s)=\widehat{\vartheta}^{i}_{m}(1)\times\widehat{\vartheta}^{i}_{m}(s,2)$
(8.5)

as in (5.24), the space $\widehat{\vartheta}^{i}_{m}(1)$ is compact and
$\widehat{\vartheta}^{i}_{m}(s,2)\equiv\vartheta^{i}_{m}(s,2)$.

######  Definition 8.2.

In section 5.2.3 (see formula (5.27)), we defined the set $W^{1}(r,m)$ as

$W^{1}(r,m):=\mathscr{F}^{1}(\mathcal{P}(r,m)\times\mathbb{R}^{m-1})\ ,$ (8.6)

where the function $\mathscr{F}^{1}$ was introduced in (5.25)-(5.26).
Similarly, for any $j\in\\{2,\dots,m\\}$, taking Remark 5.8 into account, we
had set

$\displaystyle\begin{split}W^{j}(r,m):=\mathscr{F}^{j}(\mathcal{P}(r,m)\times\mathbb{R}^{m-1})\
.\end{split}$ (8.7)

Now, by (8.3) and by the above discussion, for any $i\in\\{1,\dots,m\\}$ it
makes sense to define also

$\displaystyle\begin{split}\widehat{W}^{i}(r,m):=\mathscr{F}^{i}(\mathcal{P}(r,m)\times\overline{B}^{m-1}(0,{\mathtt{M}}(r,n,m,1)))\
.\end{split}$ (8.8)

We remind that, due to Definition 8.1 and to Remark 5.7, for any given
$i\in\\{1,\dots,m\\}$ there exists a polynomial bijection ${\mathfrak{U}}^{i}$
between $\mathcal{P}(r,m)\times\widehat{\vartheta}^{i}_{m}(1)$ and
$\widehat{W}^{i}(r,m)$: one is free to work either in the standard coordinates
$(p_{\mu},{\mathtt{a}})\in\mathbb{R}^{M}\times\mathbb{R}^{(m-1)s}$, with
$M:=\dim\mathcal{P}(r,m)$, or in the adapted coordinates
$({\mathtt{p}}_{\mu},{\mathtt{a}})\in\widehat{W}^{i}(r,m)$.

By the arguments above, without any loss of generality, for any fixed
$i\in\\{1,\dots,m\\}$ it is sufficient to consider the set of those
polynomials $P\in\mathcal{P}(r,m)$ verifying the $s$-vanishing condition on
the $s$-jet ${{\mathcal{J}}_{s,\gamma}}\in\widehat{\vartheta}^{i}_{m}(s)$ of
some curve $\gamma\in\widehat{\Theta}_{m}^{i}$. Namely, following (5.15) and
(8.1), for $i=1,\dots,m$ we introduce the semi-algebraic sets

$\displaystyle\begin{split}\widehat{Z}^{i}(r,s,m):=&\\{(P,{{\mathcal{J}}_{s,\gamma}})\in\mathcal{P}(r,m)\times\widehat{\vartheta}^{i}_{m}(s)\\}|(P,{{\mathcal{J}}_{s,\gamma}})\text{
satisfies }\\\ &\quad
q_{\ell\alpha}^{i}\circ\Phi^{i}(P,{{\mathcal{J}}_{s,\gamma}})=0\text{ for all
}\ell\in\\{1,...,m\\},\alpha\in\\{0,...,s\\}\\}\\\
\widehat{Z}(r,s,m):=&\bigcup_{i=1}^{m}\widehat{Z}^{i}(r,s,m)\\\
\widehat{\sigma}^{i}(r,s,m):=&\Pi_{\mathcal{P}(r,m)}\widehat{Z}^{i}(r,s,m)\quad,\qquad\widehat{\sigma}(r,s,m)=\bigcup_{i=1}^{m}\widehat{\sigma}^{i}(r,s,m)\\\
\widehat{\Sigma}(r,s,m):=&\bigcup_{i=1}^{n}\widehat{\Sigma}^{i}(r,s,m):=\bigcup_{i=1}^{n}\text{closure}\left(\widehat{\sigma}^{i}(r,s,m)\right)\end{split}$
(8.9)

and, of course, we have

$\widehat{Z}^{i}(r,s,m)\subset
Z^{i}(r,s,m)\quad,\qquad\widehat{\sigma}^{i}(r,s,m)\subset\sigma^{i}(r,s,m)\quad\forall\
~i\in\\{1,\dots,m\\}.$ (8.10)

For further convenience, we also state the following simple

######  Lemma 8.1.

$\widehat{\sigma}(r,s,m)$ has codimension $s+m=\text{codim }\sigma(r,s,m)$ in
$\mathcal{P}(r,m)$.

###### Proof.

By the third line of (8.9) and by the second inclusion in (8.10), it suffices
to prove the statement for $\widehat{\sigma}^{1}(r,s,m)$. By the first line of
(8.9) and by Definition 8.1, one has

$\displaystyle\begin{split}\widehat{Z}^{1}(r,s,m)=Z^{1}(r,s,m)\bigcap\left\\{(P,{{\mathcal{J}}_{s,\gamma}})\in\mathcal{P}(r,m)\times\widehat{\vartheta}^{1}_{m}(s)\right\\}\
.\end{split}$ (8.11)

As it was shown in Corollary 5.1, the Jacobian associated to the equalities
determining $Z^{1}(r,s,m)$ has full rank $ms+m$. Namely, by the discussion
below expression (5.64), in the adapted coordinates of section 5.2.3, such a
Jacobian has non-zero pivots corresponding to the derivatives w.r.t. the
coefficients of the polynomial ${\mathtt{P}}_{\mathtt{a}}$ associated to
multi-indices in the family (5.39). As we had shown in the proof of Corollary
5.1, this fact and the Implicit Function Theorem imply that for any pair
$(P,{{\mathcal{J}}_{s,\gamma}})$ belonging to $Z^{1}(r,s,m)$, one can express
$ms+m$ coefficients of $P$ as implicit functions of the other coefficients of
$P$ and of the $(m-1)s$ parameters of ${{\mathcal{J}}_{s,\gamma}}$. This was
the argument that led to estimate $\text{codim }\sigma^{1}(r,s,m)=s+m$ in
Corollary 5.1. The thesis follows by putting this argument together with
formulas (8.9)-(8.11) and with the fact that
$\dim\vartheta^{1}_{m}(s)=\dim\widehat{\vartheta}^{1}_{m}(s)$. ∎

###  8.2 Partition of $\mathcal{P}(r,m)$ and $\widehat{W}^{1}(r,m)$

Let $r,m\geq 2$ be two integers. In this paragraph, we introduce a partition
of the spaces $\mathcal{P}(r,m)$ and $\widehat{W}^{1}(r,m)$ which will turn
out to be useful in the sequel. In order to do this, we first need to
introduce a family of multi-indices.

####  8.2.1 A family of multi-indices

For $b,c\in\\{2,\dots,m\\}$, $b\leq c$ 181818We have set $b\leq c$ in (8.12)
only for convenience, in order not to have two indices $b,c$ corresponding to
the same multi-index $\mu\in{\mathbb{N}}^{m}$. Infact, it is clear that if we
eliminate this constraint we have $\varpi(b,c)=\varpi(c,b)$ for all
$b,c\in\\{2,\dots,m\\}$., we set

$\varpi(b,c):=\mu=(\mu_{1},\dots,\mu_{m})\in{\mathbb{N}}^{m}\ |\ \mu_{1}=0\ ,\
\ \mu_{j}=\delta_{jb}+\delta_{jc}\quad\forall j\in\\{2,\dots,m\\}\ .$ (8.12)

Comparing (8.12) with the sub-family $\nu({\mathtt{i}},1)$ defined in (5.39),
it is plain to check that one has the disjoint union

$\left(\bigcup_{\begin{subarray}{c}b,c=2\\\ b\leq
c\end{subarray}}^{m}\\{\varpi(b,c)\\}\right)\bigsqcup\left(\bigcup_{{\mathtt{i}}=1}^{m}\\{\nu({\mathtt{i}},1)\\}\right)=\\{\mu\in{\mathbb{N}}^{m}\
|\ |\mu|=2\\}\ .$ (8.13)

Moreover, we have the following

######  Lemma 8.2.

For any polynomial $P\in\mathcal{P}(r,m)$, the coefficients $p_{\mu}$
associated to the multi-indices $\mu$ belonging to the family (8.12) are
invariant under the transformations of paragraph 5.2.3. Namely, using the
notations in (5.20), for any given
${\mathtt{a}}\in\widehat{\vartheta}^{1}_{m}(1)$ one has

${\mathtt{p}}_{\varpi(b,c)}=p_{\varpi(b,c)}\qquad\text{ for all
}b,c\in\\{2,\dots,m\\}\quad,\qquad b\leq c\ .$

###### Proof.

We indicate by $A_{1},\dots,A_{m}$ the orthonormal basis of $\mathbb{R}^{m}$
associated to the coordinates $x_{1},\dots,x_{m}$ on which polynomials in
$\mathcal{P}(r,m)$ depend.

For any
${\mathtt{a}}:=(a_{21},\dots,a_{m1})\in\widehat{\vartheta}^{1}_{m}(1)$, we
also denote by

$v_{\mathtt{a}}:=A_{1}+a_{21}A_{2}+a_{31}A_{3}+...+a_{m1}A_{m}\ ,\ \
u_{2}:=A_{2}\ ,\ \ ...\ ,\ \ u_{m}:=A_{m}\ ,$ (8.14)

the basis associated to the adapted variables defined in Section 5.2.3 (see
(5.17)), namely

${\mathtt{y}}_{1}:=x_{1}\quad,\qquad{\mathtt{y}}_{2}={\mathtt{y}}_{2}({\mathtt{a}}):=x_{2}-a_{21}\,x_{1}\quad\dots\quad{\mathtt{y}}_{m}={\mathtt{y}}_{m}({\mathtt{a}}):=x_{m}-a_{m1}\,x_{1}\
.$ (8.15)

By (5.20), (5.39) and (8.13), the quadratic terms of the transformed
polynomial ${\mathtt{P}}_{\mathtt{a}}$ read

$\displaystyle\begin{split}\begin{cases}{\mathtt{p}}_{\nu(1,1)}{\mathtt{y}}_{1}^{2}={\mathtt{p}}_{\nu(1,1)}x_{1}^{2}\\\
{\mathtt{p}}_{\nu(\ell,1)}{\mathtt{y}}_{1}{\mathtt{y}}_{\ell}={\mathtt{p}}_{\nu(\ell,1)}x_{1}(x_{\ell}-a_{\ell
1}x_{1})\quad\ell=2,\dots,m\\\
{\mathtt{p}}_{\varpi(j,\ell)}{\mathtt{y}}_{j}{\mathtt{y}}_{\ell}={\mathtt{p}}_{\varpi(j,\ell)}(x_{j}-a_{j1}x_{1})(x_{\ell}-a_{\ell
1}x_{1})\quad j,\ell=2,\dots,m\ ,\ \ ~j\leq\ell\ .\end{cases}\end{split}$
(8.16)

By expression (8.16), we infer that - in the original variables
$x_{1},\dots,x_{m}$ \- for any $j,\ell\in\\{2,\dots,m\\}$, $j\leq\ell$, the
coefficient associated to the monomial $x_{j}x_{\ell}$ is
${\mathtt{p}}_{\varpi(j,\ell)}$, that is
$p_{\varpi(j,\ell)}={\mathtt{p}}_{\varpi(j,\ell)}\ .$

∎

####  8.2.2 Partition

For any $P\in\mathcal{P}(r,m)$, we set

$\mathbb{H}^{1}(P):=\left(\begin{matrix}2p_{\varpi(2,2)}&p_{\varpi(2,3)}&\dots&p_{\varpi(2,m)}\\\
p_{\varpi(2,3)}&2p_{\varpi(3,3)}&\dots&p_{\varpi(3,m)}\\\
\dots&\dots&\ddots&\dots\\\
p_{\varpi(2,m)}&p_{\varpi(3,m)}&\dots&2p_{\varpi(m,m)}\end{matrix}\right)$
(8.17)

and

${\mathsf{S}}_{1}^{1}(r,m):=\\{P\in\mathcal{P}(r,m)|\
\det\mathbb{H}^{1}(P)\neq 0\\}\ .$ (8.18)

######  Remark 8.2.

Matrix $\mathbb{H}^{1}(P)$ is invariant under the transformations of section
5.2.3. Namely, by Lemma 8.2, we have

$\mathbb{H}^{1}({\mathtt{P}}_{\mathtt{a}}):=\left(\begin{matrix}2{\mathtt{p}}_{\varpi(2,2)}&{\mathtt{p}}_{\varpi(2,3)}&\dots&{\mathtt{p}}_{\varpi(2,m)}\\\
{\mathtt{p}}_{\varpi(2,3)}&2{\mathtt{p}}_{\varpi(3,3)}&\dots&{\mathtt{p}}_{\varpi(3,m)}\\\
\dots&\dots&\ddots&\dots\\\
{\mathtt{p}}_{\varpi(2,m)}&{\mathtt{p}}_{\varpi(3,m)}&\dots&2{\mathtt{p}}_{\varpi(m,m)}\end{matrix}\right)=\mathbb{H}^{1}(P)\
.$ (8.19)

We also define

${\mathsf{S}}_{2}^{1}(r,m):=\mathcal{P}(r,m)\backslash{\mathsf{S}}_{1}^{1}(r,m)=\\{P\in\mathcal{P}(r,m)|\
\det\mathbb{H}^{1}(P)=0\\}\ ,$ (8.20)

so that we have the disjoint union

$\mathcal{P}(r,m)={\mathsf{S}}_{1}^{1}(r,m)\bigsqcup{\mathsf{S}}_{2}^{1}(r,m)\
.$ (8.21)

We now consider the images of ${\mathsf{S}}_{1}^{1}(r,m)$ and
${\mathsf{S}}_{2}^{1}(r,m)$ through the transformation ${\mathfrak{U}}^{1}$
defined in Remark 5.7, namely

$\displaystyle\begin{split}{\mathscr{S}}^{1}_{1}(r,m):={\mathfrak{U}}^{1}\left({\mathsf{S}}_{1}^{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(1)\right)=&\left\\{({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}})\in\widehat{W}^{1}(r,m)|\det\mathbb{H}^{1}({\mathtt{P}}_{\mathtt{a}})\neq
0\right\\}\\\
{\mathscr{S}}^{1}_{2}(r,m):={\mathfrak{U}}^{1}\left({\mathsf{S}}_{2}^{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(1)\right)=&\left\\{({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}})\in\widehat{W}^{1}(r,m)\right.|\det\mathbb{H}^{1}({\mathtt{P}}_{\mathtt{a}})=0\Big{\\}}\
.\end{split}$ (8.22)

By (8.22), we have the partition

$\widehat{W}^{1}(r,m)={\mathscr{S}}^{1}_{1}(r,m)\bigsqcup{\mathscr{S}}^{1}_{2}(r,m)\
.$ (8.23)

######  Remark 8.3.

It is clear that the above partition can be implemented also in case one
considers adapted variables
$({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}})\in\widehat{W}^{i}(r,m)$, with
$i\in\\{2,\dots,m\\}$. By suitably modifying the family of indices in (8.12),
as well as by introducing an adapted matrix $\mathbb{H}^{i}(P)$, it is
possible to define sets ${\mathsf{S}}_{1}^{i},{\mathsf{S}}_{2}^{i}$ whose
disjoint union yields $\mathcal{P}(r,m)$ and sets
${\mathscr{S}}_{1}^{i},{\mathscr{S}}_{2}^{i}$ whose disjoint union yields
$\widehat{W}^{i}(r,m)$. However, the underlying reasonings are not
conceptually different from the ones we did above, therefore we omit them in
order not to burden the exposition.

###  8.3 Two important results

Consider three integers $r,m\geq 2$ and $s\in\\{1,\dots,r-1\\}$. The two
results below are the cornerstones of the proof of Theorems C1-C2-C3.

######  Theorem 8.1.

In case $r\geq 2,s=1$, for any $i\in\\{1,\dots,m\\}$ the semi-algebraic sets
$\widehat{\sigma}^{i}(r,1,m)$, and $\widehat{\sigma}(r,1,m)$ are closed in
$\mathcal{P}(r,m)$, that is, by formulas (8.9),

$\widehat{\sigma}^{i}(r,1,m)=\widehat{\Sigma}^{i}(r,1,m)\quad\forall\
i\in\\{1,\dots,m\\}\quad,\qquad\widehat{\sigma}(r,1,m)=\widehat{\Sigma}(r,1,m)\
.$

Moreover, for any $i\in\\{1,\dots,m\\}$, taking the definition of
transformation $\Upsilon^{i}$ into account (see (5.29)), the set
$\Pi_{\widehat{W}^{i}(r,m)}\Upsilon^{i}(\widehat{Z}^{i}(r,1,m))$ is closed in
$\widehat{W}^{i}(r,m)$, and its form can be explicitly computed.

######  Theorem 8.2.

For any given values of $r\geq 3$, $s\geq 2$, and $i\in\\{1,\dots,m\\}$ there
exist two semi-algebraic subsets of $\mathcal{P}(r,m)$

$\displaystyle\begin{split}&X_{1}^{i}(r,s,m)\subset{\mathsf{S}}_{1}^{i}(r,m)\
,\ \ X_{2}^{i}(r,s,m)\subset{\mathsf{S}}_{2}^{i}(r,m)\ ,\end{split}$ (8.24)

and two semi-algebraic subsets of $\widehat{W}^{i}(r,m)$

$\displaystyle\begin{split}&Y_{1}^{i}(r,s,m)\subset{\mathscr{S}}_{1}^{i}(r,m)\
,\ \ Y_{2}^{i}(r,s,m)\subset{\mathscr{S}}_{2}^{i}(r,m)\ ,\ \end{split}$ (8.25)

verifying the following properties:

  1. 1.

one has the partition $\widehat{\sigma}^{i}(r,s,m)=X_{1}^{i}(r,s,m)\bigsqcup
X_{2}^{i}(r,s,m)\ ;$

  2. 2.

for $j\in\\{1,2\\}$, one has

$X_{j}^{i}(r,s,m)=\Pi_{\mathcal{P}(r,m)}\left(({\mathfrak{U}}^{i})^{-1}(Y_{j}^{i}(r,s,m))\right)\
,$

where ${\mathfrak{U}}^{i}$ was defined in Remarks 5.7-5.8;

  3. 3.

$Y_{1}^{i}(r,s,m)$ is closed in ${\mathscr{S}}_{1}^{i}(r,m)$ for the topology
induced by $\mathcal{P}(r,m)$;

  4. 4.

$X_{1}^{i}(r,s,m)$ is closed in ${\mathsf{S}}_{1}^{i}(r,m)$ for the topology
induced by $\mathcal{P}(r,m)$;

  5. 5.

the form of $Y_{1}^{i}(r,s,m)$ can be explicitly computed starting from the
expression of $\widehat{Z}^{i}(r,s,m)$ by the means of an algorithm involving
only linear operations.

The rest of this section is devoted to the proof of the above results.

We will only prove Theorems 8.1-8.2 in the case $i=1$, as the other cases are
simple generalizations.

####  8.3.1 Proof of Theorem 8.1 and strategy of proof of Theorem 8.2.

We have seen in section 5 that the equations determining $Z^{1}(r,s,m)$, can
be written in the adapted coordinates introduced in section 5.2.3. Namely, by
recalling the functions

${\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}):W^{1}(r,m)\times\mathbb{R}^{(m-1)(s-1)}\longrightarrow\mathbb{R}\quad,\qquad{\mathtt{i}}\in\\{1,\dots,m\\}\quad\alpha\in\\{0,\dots,s\\}\
,$ (8.26)

presented in (5.36)-(5.37), Theorem 5.1 ensures that $Z^{1}(r,s,m)$ is the
image through the inverse of the transformation $\Upsilon^{1}$ in (5.30) of
the zero set

$\bigcap_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\\{({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})\in
W^{1}(r,m)\times\mathbb{R}^{(m-1)(s-1)}|{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0\\}\
,$ (8.27)

and the explicit form of the quantities ${\mathtt{Q}}_{i\alpha}$ is given in
(5.43)-(5.44).

Then, if we indicate by $\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}$ the
restriction of ${\mathtt{Q}}_{{\mathtt{i}}\alpha}$ to the subset
$\widehat{W}^{1}(r,m)\times\vartheta^{1}_{m}(s,2)\subset
W^{1}(r,m)\times\vartheta^{1}_{m}(s,2)$, and if we denote by
${\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\right)$ the zero set of the non-linear
maps
$\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}$ , it is clear by the discussion at
paragraph 8.1, in particular by (8.11), that

$\displaystyle\begin{split}\widehat{Z}^{1}(r,s,m):=(\Upsilon^{1})^{-1}\left({\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\right)\right)\\\ \end{split}$ (8.28)

that is

$\displaystyle\begin{split}\Upsilon^{1}(\widehat{Z}^{1}(r,s,m))={\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\right)\ .\end{split}$ (8.29)

###### Proof.

(Theorem 8.1) When $s=1$, then in (5.43)-(5.44) one must consider
$\alpha\in\\{0,1\\}$. We observe that

$\forall\ \alpha\in\\{0,1\\}\ ,\ \
~~~\forall{\mathtt{i}}=1,\dots,m\qquad\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0\quad\Longleftrightarrow\quad{\mathtt{p}}_{\nu({\mathtt{i}},\alpha)}=0\
,$ (8.30)

where the family of multi-indices $\nu({\mathtt{i}},\alpha)$ was defined in
(5.39). As we see, no parameters belonging to the space
$\vartheta^{1}_{m}(s,2)$ appear in equation (8.30), which determines a closed
subset of $\widehat{W}^{1}(r,m)$. In other words, by (8.29) with $s=1$, we
have that the projection
$\Pi_{\widehat{W}^{1}(r,m)}\left(\Upsilon^{1}(\widehat{Z}^{1}(r,1,m))\right)$
is closed in $\widehat{W}^{1}(r,m)$ and its form is given in (8.30). Moreover,
as the function ${\mathfrak{U}}^{1}$ introduced in Remark 5.7 is continuous by
construction, the inverse image
$({\mathfrak{U}}^{1})^{-1}\left(\Pi_{\widehat{W}^{1}(r,m)}\left(\Upsilon^{1}(\widehat{Z}^{1}(r,1,m))\right)\right)$
is closed in $\mathcal{P}(r,n)\times\widehat{\vartheta}^{1}_{m}(1)$. Finally,
as all the elements of $\widehat{\vartheta}^{1}_{m}(1)$ live in a compact ball
(see (8.3) and Remark (8.1)), using Lemma E.1 with $E\equiv\mathcal{P}(r,m)$
and $K\equiv\widehat{\vartheta}^{1}_{m}(1)$ and taking formulas (8.9)-(8.29)
into account one has that

$\widehat{\sigma}^{1}(r,1,m)=\Pi_{\mathcal{P}(r,m)}\left[({\mathfrak{U}}^{1})^{-1}\left(\Pi_{\widehat{W}^{1}(r,m)}\left(\Upsilon^{1}(\widehat{Z}^{1}(r,1,m))\right)\right)\right]$
(8.31)

is closed. This concludes the proof. ∎

Now, taking (8.29), the third line of (8.9) and the proof of Theorem 8.1 into
account, the key idea behind the proof of Theorem 8.2 consists in
understanding under which conditions the set
$\Upsilon^{1}(\widehat{Z}^{1}(r,s,m))$ admits a closed projection onto
$\widehat{W}^{1}(r,m)$.

In particular, we will see that if we only consider polynomials in the set
${\mathsf{S}}^{1}_{1}(r,m)\subset\mathcal{P}(r,m)$, it is possible to
extrapolate linearly from the $ms+m$ equations in (8.29) the $(m-1)s$
parameters of $\vartheta^{1}_{m}(s,2)$ as explicit algebraic functions of the
parameters of
${\mathscr{S}}^{1}_{1}(r,m):={\mathfrak{U}}^{1}({\mathsf{S}}^{1}_{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(1))$,
as the discussion at the following paragraph shows.

We start by setting

$\displaystyle\begin{split}{\mathsf{A}}(r,s,m):=&{\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\right)\bigcap\Upsilon^{1}({\mathsf{S}}_{1}^{1}(r,s,m)\times\widehat{\vartheta}^{1}_{m}(s))\
,\end{split}$ (8.32)

and we remind that, by Remark 5.7,

$\Upsilon^{1}({\mathsf{S}}_{1}^{1}(r,s,m)\times\widehat{\vartheta}^{1}_{m}(s))={\mathfrak{U}}^{1}\left({\mathsf{S}}_{1}^{1}(r,s,m)\times\widehat{\vartheta}^{1}_{m}(1)\right)\times\vartheta^{1}_{m}(s,2)={\mathscr{S}}^{1}_{1}(r,m)\times\vartheta^{1}_{m}(s,2)\
.$ (8.33)

We now claim that ${\mathsf{A}}(r,s,m)$ is a graph of the form

$\displaystyle\begin{split}{\mathsf{A}}(r,s,m)=Y^{1}_{1}(r,s,m)\times\overline{\theta}(r,s,m)\
,\end{split}$ (8.34)

where

$Y^{1}_{1}(r,s,m):=\Pi_{\widehat{W}^{1}(r,m)}{\mathsf{A}}(r,s,m)$

is a closed subset of ${\mathscr{S}}^{1}_{1}(r,m)$ for the induced topology
determined by algebraic equations involving the coordinates of elements in
${\mathscr{S}}^{1}_{1}(r,m)$, and the points of
$\overline{\theta}(r,s,m)\subset\vartheta^{1}_{m}(s,2)$ are parametrized by
$Y^{1}_{1}(r,s,m)$. In the following subparagraphs, we will prove this claim
and show how it can be used in order to demonstrate Theorem 8.2.

####  8.3.2 Explicit form of the equations (case $r\geq 3$, $s\geq 2$)

The goal of this paragraph is to give a more explicit expression of
${\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,1\\}\end{subarray}}\right)$, in case $r\geq 3$, $s\geq 2$.

######  Remark 8.4.

We remind that equations (5.43)-(5.44) are recursive w.r.t. the parameters of
the curve $\gamma$. Namely, for any given integer $\beta\in\\{2,\dots,s\\}$,
the coefficients of order $\beta$ belonging to the space
$\vartheta^{1}_{m}(s,2)$ \- that is
$a_{22},\dots,a_{2\beta},\dots,a_{m2},\dots,a_{m\beta}$ \- appear in equations
(5.43)-(5.44) only for $\alpha\geq\beta$.

For any polynomial $P\in\mathcal{P}(r,m)$ and any curve
$\gamma\in\widehat{\Theta}_{m}^{1}$, we indicate by
${\mathtt{P}}_{\mathtt{a}}^{(>2)}$ the associated polynomial
${\mathtt{P}}_{\mathtt{a}}$ written in the adapted coordinates for $\gamma$
(see paragraph 5.2.3) deprived of its monomials of degree less or equal than
two 191919One has ${\mathtt{P}}_{\mathtt{a}}^{(>2)}\neq 0$ since we are
considering the case of polynomials having degree $r\geq 3$.. Also, if
${{\mathcal{J}}_{s,\gamma}}\in\widehat{\vartheta}^{1}_{m}(1)$ is the
$s$-truncation of $\gamma$, for any given $\alpha\in\\{2,\dots,s\\}$ we denote
by

${\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}(t):={\mathtt{J}_{s,\gamma,{\mathtt{a}}}}-\left(\begin{matrix}0\\\
\sum_{i=\alpha}^{s}a_{2i}t^{i}\\\ \dots\\\ \sum_{j=\alpha}^{s}a_{mj}t^{j}\\\
\end{matrix}\right)$

its truncation at order $\alpha-1$ written in the adapted coordinates for
$\gamma$.

######  Remark 8.5.

We observe that, for $\alpha=2$,
${\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<2)}(t)$ reduces to the line
$(t,0,\dots,0)$, since with the exception of the parametrizing coordinate, the
components of ${\mathtt{J}_{s,\gamma,{\mathtt{a}}}}$ start at order two in $t$
(see paragraph 5.2.3).

With this setting, we have

######  Lemma 8.3.

For any polynomial $P\in\mathcal{P}(r,m)$, there exists a linear change of
coordinates
${\mathfrak{D}}={\mathfrak{D}}(P):\mathbb{R}^{m}\longrightarrow\mathbb{R}^{m}$
such that for any given $\alpha\in\\{2,\dots,s\\}$, and for
${\mathtt{i}}=1,\dots,m$, the algebraic equations
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
in (5.43)-(5.44) take the form

$\left(\begin{matrix}0&0&0&\dots&0\\\
0&2{\mathtt{p}}^{\prime}_{\varpi(2,2)}&0&\dots&0\\\
0&0&2{\mathtt{p}}^{\prime}_{\varpi(3,3)}&\dots&0\\\
0&\dots&\dots&\dots&\dots\\\
0&0&0&\dots&2{\mathtt{p}}^{\prime}_{\varpi(m,m)}\\\
\end{matrix}\right)\left(\begin{matrix}0\\\ a^{\prime}_{2\alpha}\\\
a^{\prime}_{3\alpha}\\\ \dots\\\
a^{\prime}_{m\alpha}\end{matrix}\right)+{\mathfrak{D}}\left(\begin{matrix}\widehat{\mathtt{Q}}_{1\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\widehat{\mathtt{Q}}_{2\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\widehat{\mathtt{Q}}_{3\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\dots\\\
\widehat{\mathtt{Q}}_{m\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\end{matrix}\right)=0\
.$ (8.35)

###### Proof.

Step 1. We firstly claim that equations (5.43)-(5.44) can be but into the form

$\left(\begin{matrix}0&0&0&\dots&0\\\
0&2{\mathtt{p}}_{\varpi(2,2)}&{\mathtt{p}}_{\varpi(2,3)}&\dots&{\mathtt{p}}_{\varpi(2,m)}\\\
0&{\mathtt{p}}_{\varpi(2,3)}&2{\mathtt{p}}_{\varpi(3,3)}&\dots&{\mathtt{p}}_{\varpi(3,m)}\\\
0&\dots&\dots&\dots&\dots\\\
0&{\mathtt{p}}_{\varpi(2,m)}&{\mathtt{p}}_{\varpi(3,m)}&\dots&2{\mathtt{p}}_{\varpi(m,m)}\\\
\end{matrix}\right)\left(\begin{matrix}0\\\ a_{2\alpha}\\\ a_{3\alpha}\\\
\dots\\\
a_{m\alpha}\end{matrix}\right)+\left(\begin{matrix}\widehat{\mathtt{Q}}_{1\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\widehat{\mathtt{Q}}_{2\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\widehat{\mathtt{Q}}_{3\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\dots\\\
\widehat{\mathtt{Q}}_{m\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\end{matrix}\right)=0\
.$ (8.36)

By (8.30), for any $\alpha\in\\{2,\dots,s\\}$, and for any
${\mathtt{i}},{\mathtt{j}}\in\\{1,\dots,m\\}$, the monomials
${\mathtt{p}}_{\nu({\mathtt{j}},0)},{\mathtt{p}}_{\nu({\mathtt{j}},1)}$ do not
yield any contribution to equations
${\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$.
Therefore, taking (8.13) into account, the only monomials of order two which
may appear in equation
${\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
are those associated to the multi-indices $\varpi(b,c)$, with
$b,c\in\\{2,\dots,m\\}$, $b\leq c$.

Moreover, fixing the values of ${\mathtt{i}}\in\\{1,\dots,m\\}$ and
$\alpha\in\\{2,\dots,s\\}$, by (5.43)-(5.44), the multi-indices
$\mu\in{\mathbb{N}}^{m}$ contributing to equation
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
are those for which the set
${\mathcal{G}}_{m}(\widetilde{\mu}({\mathtt{i}}),\alpha)$ in (5.41) is non-
empty. This amounts to requiring that the components of the vector
$(k_{22},\dots,k_{2\alpha},\dots,k_{m2},\dots,k_{m\alpha})\in{\mathbb{N}}^{(m-1)(\alpha-1)}$
appearing in formulas (5.43)-(5.44) satisfy

$\sum_{i=2}^{\alpha}k_{ji}=\widetilde{\mu}_{j}({\mathtt{i}})\quad\forall
j\in\\{2,\dots,m\\}\quad,\qquad\widetilde{\mu}_{1}({\mathtt{i}})+\sum_{j=2}^{m}\sum_{i=2}^{\alpha}i\
k_{ji}=\alpha\ .$ (8.37)

In particular, for fixed ${\mathtt{i}}\in\\{1,\dots,m\\}$,
$\alpha\in\\{2,\dots,s\\}$ and $\ell\in\\{2,\dots,m\\}$, if we look at the
monomials containing the coefficient $a_{\ell\alpha}$ in equation
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
\- that is, at the form of the terms for which $k_{\ell\alpha}\neq 0$ in
(5.43)-(5.44) - by (8.37) we must have

$\displaystyle\begin{split}\widetilde{\mu}_{1}({\mathtt{i}})=0\quad,\qquad
k_{ji}=\delta_{i\alpha}\delta_{j\ell}\quad,\qquad\widetilde{\mu}_{j}({\mathtt{i}})=\delta_{j\ell}\quad,\qquad
j\in\\{2,\dots,m\\}\ .\end{split}$ (8.38)

Firstly, expression (8.38) implies $|\mu|=2$. Secondly, as we said above, the
only multi-indices of length $|\mu|=2$ which may yield a contribution to
${\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
are those belonging to the family
$\\{\varpi(b,c)\\}_{\begin{subarray}{c}b,c\in\\{2,\dots,m\\}b\leq
c\end{subarray}}$ in (8.12). Therefore, we have two cases.

Case ${\mathtt{i}}=1$. If
$\mu\in\\{\varpi(b,\ell)\\}_{\begin{subarray}{c}b,\ell\in\\{2,\dots,m\\}b\leq\ell\end{subarray}}$
then $\mu_{1}=0$, and all terms in formula (5.43) which are associated to
these indices are multiplied by zero. Hence, the coefficients $a_{\ell\alpha}$
do not appear in
$\widehat{\mathtt{Q}}_{1\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$,
nor do any of the monomials of order two in ${\mathtt{P}}_{\mathtt{a}}$. This,
together with Remark 8.4 proves the claim for ${\mathtt{i}}=1$ (the first line
of (8.36)).

Case ${\mathtt{i}}\in\\{2,\dots,m\\}$. Taking (8.38) into account, for any
given $\ell\in\\{2,\dots,m\\}$ one has that

  1. 1.

if ${\mathtt{i}}\leq\ell$, the only term to which the coefficient
$a_{\ell\alpha}$ is associated in equation
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
is the one corresponding to the multi-index $\mu\in\varpi({\mathtt{i}},\ell)$,
that is, by (5.44), the monomial
$(1+\delta_{\ell{\mathtt{i}}}){\mathtt{p}}_{\varpi({\mathtt{i}},\ell)}a_{\ell\alpha}$;

  2. 2.

if ${\mathtt{i}}>\ell$, the term containing $a_{\ell\alpha}$ is the one
associated to the multi-index $\mu\in\varpi(\ell,{\mathtt{i}})$, that is, by
(5.44),
$(1+\delta_{\ell{\mathtt{i}}}){\mathtt{p}}_{\varpi(\ell,{\mathtt{i}})}a_{\ell\alpha}$.

Conversely, if a monomial associated to an index $\varpi({\mathtt{i}},\ell)$,
with ${\mathtt{i}},\ell\in\\{2,\dots,m\\}$, ${\mathtt{i}}\leq\ell$, appears in
equations
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$,
then, by (8.12) and by (8.37), one must necessarily have

$\displaystyle\begin{cases}\widetilde{\mu}_{1}({\mathtt{i}})+\sum_{j=2}^{m}\sum_{i=2}^{\alpha}i\
k_{ji}=\alpha\quad,\qquad\widetilde{\mu}_{1}({\mathtt{i}})=0\\\
\sum_{i=2}^{\alpha}k_{ji}=\widetilde{\mu}_{j}({\mathtt{i}}):=\delta_{j\ell}+\delta_{j{\mathtt{i}}}-\delta_{j{\mathtt{i}}}=\delta_{j\ell}\quad\forall
j\in\\{2,\dots,m\\}\ ,\end{cases}$ (8.39)

which is true if and only if for some $v\in\\{2,\dots,\alpha\\}$ one has

$\displaystyle\begin{cases}k_{ji}=\delta_{j\ell}\delta_{iv}\\\
\sum_{j=2}^{m}\sum_{i=2}^{\alpha}i\
\delta_{j\ell}\delta_{iv}=\alpha\end{cases}$ (8.40)

that is if and only if $k_{ji}=\delta_{j\ell}\delta_{i\alpha}$ . One can check
by formula (5.44) that this ensures that such a term must be of the form
$(1+\delta_{\ell{\mathtt{i}}}){\mathtt{p}}_{\varpi({\mathtt{i}},\ell)}a_{\ell\alpha}$.
This reasoning, together with Remark 8.4, and with the fact that - as we
showed at the beginning of the proof - no monomials of order two appear in
equation
$\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})=0$
other than those associated to the family (8.12), proves the claim for
${\mathtt{i}}\in\\{2,\dots,m\\}$ (rows $2,\dots,m$ of (8.36)).

Step 2. For any $P\in\mathcal{P}(r,m)$, taking Lemma 8.2 into account we
indicate by $\mathbb{G}^{1}(P)$ the matrix

$\left(\begin{matrix}0&0&\dots&0\\\
0&2p_{\varpi(2,2)}&\dots&p_{\varpi(2,m)}\\\ 0&\dots&\ddots&\dots\\\
0&p_{\varpi(2,m)}&\dots&2p_{\varpi(m,m)}\\\
\end{matrix}\right)=\left(\begin{matrix}0&0&\dots&0\\\
0&2{\mathtt{p}}_{\varpi(2,2)}&\dots&{\mathtt{p}}_{\varpi(2,m)}\\\
0&\dots&\ddots&\dots\\\
0&{\mathtt{p}}_{\varpi(2,m)}&\dots&2{\mathtt{p}}_{\varpi(m,m)}\\\
\end{matrix}\right)=\left(\begin{matrix}0&0\\\
0&\mathbb{H}^{1}(P)\end{matrix}\right)$ (8.41)

appearing in (8.36). $\mathbb{G}^{1}(P)$ is symmetric, hence diagonalizable.
Hence, for any $P\in\mathcal{P}(r,m)$, there exist a basis of eigenvectors,
indicated by

$v_{\mathtt{a}},\,u^{\prime}_{2}={\mathtt{u}}^{\prime}_{2}(P),\dots,\,u^{\prime}_{m}=u^{\prime}_{m}(P)\
,$ (8.42)

and a real $m\times m$ invertible matrix ${\mathfrak{D}}={\mathfrak{D}}(P)$,
such that equation (8.35) takes the form

$\left(\begin{matrix}0&0&0&\dots&0\\\
0&2{\mathtt{p}}^{\prime}_{\varpi(2,2)}&0&\dots&0\\\
0&0&2{\mathtt{p}}_{\varpi(3,3)}^{\prime}&\dots&0\\\
0&\dots&\dots&\dots&\dots\\\
0&0&0&\dots&2{\mathtt{p}}^{\prime}_{\varpi(m,m)}\\\
\end{matrix}\right)\left(\begin{matrix}0\\\ a^{\prime}_{2\alpha}\\\
a^{\prime}_{3\alpha}\\\ \dots\\\
a^{\prime}_{m\alpha}\end{matrix}\right)+{\mathfrak{D}}\left(\begin{matrix}{\mathtt{Q}}_{1\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
{\mathtt{Q}}_{2\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
{\mathtt{Q}}_{3\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\\\
\dots\\\
{\mathtt{Q}}_{m\alpha}\left({\mathtt{P}}_{\mathtt{a}}^{(>2)},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}}^{(<\alpha)}\right)\end{matrix}\right)=0\
,$ (8.43)

where the primed quantities indicate that we are working in the new basis
(8.42).

######  Remark 8.6.

Comparing (8.14) with (8.42), we observe that the vector $v_{\mathtt{a}}$ was
left unchanged. This is due to the fact that, by (8.41), $v_{\mathtt{a}}$ is
already an eigenvector of $\mathbb{G}^{1}(P)$ (associated to a null
eigenvalue).

∎

####  8.3.3 Proof of Theorem 8.2

For $r\geq 3$, $s\geq 2$, we define

$X_{1}^{1}(r,s,m):=\widehat{\sigma}^{1}(r,s,m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)\
.$ (8.44)

Step 1. Firstly, we show that, if $P\in{\mathsf{S}}_{1}^{1}(r,m)$, then the
parameters of the space $\vartheta^{1}_{m}(s,2)$ can be reduced iteratively
from equation (8.35) for $\alpha\in\\{2,\dots,s\\}$.

When $\alpha=2$ the second term at the l.h.s. of formula (8.35) in Lemma 8.3.2
does not depend on the parameters of $\vartheta^{1}_{m}(s,2)$ (see Remark
8.5), so that the coefficients $a_{22},\dots,a_{m2}$ can be reduced, as matrix
$\mathbb{H}^{1}(P)$ is invertible by construction when
$P\in{\mathsf{S}}_{1}^{1}(r,m)$.

If, for $\alpha\in\\{3,\dots,s\\}$, we assume that the parameters
$a_{j\beta}$, with $j\in\\{2,\dots,m\\}$ and $\beta\in\\{2,\dots,\alpha-1\\}$,
have been reduced, then the first equation in (8.35) does not contain any new
parameter, whereas the terms $a_{2\alpha},\dots,a_{m\alpha}$ can be found by
inverting $\mathbb{H}^{1}(P)$ once again.

The above considerations and (8.44) imply that if $P\in X_{1}^{1}(r,s,m)$ then

  1. 1.

the parameters of the space $\vartheta^{1}_{m}(s,2)$ can be reduced from the
equations in (8.35) by the means of a recursive algorithm which only involves
linear computations and the inversion of $\mathbb{H}^{1}(P)$;

  2. 2.

there exists a truncation ${{\mathcal{J}}_{s,\gamma}}\in\vartheta^{1}_{m}(s)$
such that
${\mathfrak{U}}^{1}(P\times{{\mathcal{J}}_{s,\gamma}})=({\mathtt{P}}_{\mathtt{a}},{\mathtt{a}},{\mathtt{J}_{s,\gamma,{\mathtt{a}}}})$
solves (8.35).

Taking (8.32) and the above arguments into account, with the notations of
section 8.3.1 we have that ${\mathsf{A}}(r,s,m)$ is determined by a system of
$ms+m-(m-1)(s-1)=s+2m-1$ explicit algebraic equations involving only the
coordinates of ${\mathscr{S}}_{1}^{1}(r,m)$, and of $(m-1)(s-1)$ explicit
algebraic equations that parametrize the coefficients of
$\vartheta^{1}_{m}(s,2)$ as functions of the points in
${\mathscr{S}}_{1}^{1}(r,m)$. In other words, ${\mathsf{A}}(r,s,m)$ has the
form of a graph of the type indicated in (8.34), and $Y^{1}_{1}(r,s,m)$ is a
closed algebraic subset of ${\mathscr{S}}^{1}_{1}(r,m)$ whose form can be
explicitly computed - starting from the expression of
$\Upsilon^{1}(\widehat{Z}^{1}(r,s,m))$ \- by the means of an algorithm
involving only linear operations.

Moreover, by Remark 5.7 the function ${\mathfrak{U}}^{1}$ is polynomial, and
${\mathsf{S}}_{1}^{1}(r,s,m)$ is obviously semi-algebraic in
$\mathcal{P}(r,m)$ by construction. Therefore, by (8.32)-(8.33) the set
${\mathsf{A}}(r,s,m)$ is semi-algebraic in
$\widehat{W}^{1}(r,m)\times\vartheta^{1}_{m}(s,2)$, and $Y^{1}_{1}(r,s,m)$ is
semi-algebraic in $\widehat{W}^{1}(r,m)$ by the Theorem of Tarski and
Seidenberg A.1.

Step 2. Since the invertible transformation ${\mathfrak{U}}^{1}$ defined in
Remark 5.7 is polynomial, due to Step 1 (see (8.32)-(8.33)-(8.34)) and to
continuity we have that the inverse image

$({\mathfrak{U}}^{1})^{-1}(Y^{1}_{1}(r,s,m))\subset{\mathsf{S}}_{1}^{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(1)$
(8.45)

is closed in ${\mathsf{S}}_{1}^{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(1)$
for the induced topology. Finally by taking (8.32)-(8.34)-(8.44) into account,
and by considering the fact that, as we have already pointed out in Remark
8.1, $\widehat{\vartheta}^{1}_{m}(1)$ is compact - Lemma E.1 ensures that the
projection

$\Pi_{\mathcal{P}(r,m)}\left(({\mathfrak{U}}^{1})^{-1}(Y^{1}_{1}(r,s,m))\right)=X^{1}_{1}(r,s,m)=\widehat{\sigma}^{1}(r,s,m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)$
(8.46)

is closed in ${\mathsf{S}}_{1}^{1}(r,s,m)$ for the topology induced by
$\mathcal{P}(r,m)$.

The semi-algebraicness of the projection in (8.46) is a consequence of the
semi-algebraicness of $Y^{1}_{1}(r,s,m)$ demonstrated at Step 1, of the fact
that ${\mathfrak{U}}^{1}$ is polynomial, and of the Theorem of Tarski and
Seidenberg.

The above arguments prove Theorem 8.2 once one sets

$\begin{split}&X_{2}^{1}(r,s,m):=\widehat{\sigma}^{1}(r,s,m)\backslash
X_{1}^{1}(r,s,m)\\\
~&Y_{2}^{1}(r,s,m):=\Pi_{\widehat{W}^{1}(r,m)}\left({\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s\\}\end{subarray}}\right)\right)\backslash
Y_{1}^{1}(r,s,m)\ .\end{split}$ (8.47)

##  9 Proof of Theorems C1-C2-C3

We assume the notations of section 8. In this section, we consider two
positive integers $r\geq 2$, $n\geq 3$, a vector
${\mathtt{s}}:=(s_{1},\dots,s_{n-1})\in{\mathbb{N}}^{n-1}$, with $1\leq
s_{i}\leq r-1$ for all $i=1,\dots,n-1$, and a function $h$ of class
$C_{b}^{2r-1}$ around the origin, verifying $\nabla h(0)\neq 0$.

Also, for any $m\in\\{2,\dots,n-1\\}$, we set
${\mathscr{K}}(r,n,m):={\mathtt{M}}(r,n,m,1)$, where the constant
${\mathtt{M}}(r,n,m,1)$ is the one appearing in the statement of Theorem 4.1
(see formula (4.1)).

###  9.1 Proof of Theorem C1

Fix $m\in\\{2,\dots,n-1\\}$. Let $\Gamma^{m}$ be a given $m$-dimensional
subspace belonging to the subset
${\mathsf{\Lambda}}_{0}(h,m,n)\subset{\mathtt{G}}(m,n)$ introduced in
Definition 3.2.

Taking Definition 8.3 into account, we consider a curve
$\gamma\in\widehat{\Theta}_{m}$ whose image is contained in $\Gamma^{m}$.
Without any loss of generality, up to changing the order of the vectors
spanning $\Gamma^{m}$, we can suppose that $\gamma$ is parametrized by the
first coordinate, hence that $\gamma\in\widehat{\Theta}_{m}^{1}$. Following
(8.2), we indicate by
${\mathtt{a}}=(a_{21},\dots,a_{m1})\in\overline{B}^{m-1}(0,{\mathscr{K}}(r,n,m))$
the linear coefficients coefficients of the Taylor expansion of $\gamma$ at
the origin, and by ${{\mathcal{J}}_{s,\gamma}}$ its $s$-truncation (with
$1\leq s\leq r-1$).

We also indicate by $u_{1},\dots,u_{m}\in{\mathtt{U}}(m,n)$ a orthonormal
basis spanning $\Gamma^{m}$, and by
$v:=u_{1}+\sum_{i=2}^{m}a_{i1}u_{i},u_{2}\dots,u_{m}$ the basis associated to
the adapted coordinates for $\gamma$ introduced in section 5.2.3. As we have
already shown in (7.3), the Taylor polynomial ${\mathtt{T}}_{0}(h,r,n)$
restricted to $\Gamma^{m}$ written in the adapted coordinates reads

${\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m)({\mathtt{y}})=\sum_{\begin{subarray}{c}\mu\in{\mathbb{N}}^{m}\\\
1\leq|\mu|\leq
r\end{subarray}}\frac{1}{\mu!}h_{0}^{|\mu|}\Big{[}\stackrel{{\scriptstyle\mu_{1}}}{{\overbrace{v}}},\stackrel{{\scriptstyle\mu_{2}}}{{\overbrace{u_{2}}}},\dots,\stackrel{{\scriptstyle\mu_{m}}}{{\overbrace{u_{m}}}}\Big{]}\,{\mathtt{y}}_{1}^{\mu_{1}}\dots\mathtt{y}_{m}^{\mu_{m}}\
,$ (9.1)

where we have used the notation introduced in formula (2.2). Moreover, as
customary, ${{\mathcal{J}}_{s,\gamma}}$ reads
${\mathtt{J}_{s,\gamma,{\mathtt{a}}}}$ in the new coordinates.

By the arguments at paragraph 8.1, and by taking (5.29) and (8.9) into
account, if we manage to prove that condition

$({\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m),{\mathtt{a}},{{\mathtt{J}}_{1,\gamma,{\mathtt{a}}}})\in\Upsilon^{1}(\widehat{Z}^{1}(r,1,m))$

is never satisfied for any choice of the subspace
$\Gamma^{m}\in{\mathsf{\Lambda}}_{0}(h,m,n)$ and of the curve $\gamma$, which
is equivalent - due to Theorem 8.1 for $s=1$ \- to condition

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\not\in\widehat{\sigma}^{1}(r,1,m)=\widehat{\Sigma}^{1}(r,1,m)\quad,\qquad\forall\
\Gamma^{m}\in{\mathsf{\Lambda}}_{0}(h,m,n)\ ,$ (9.2)

then by the definitions in (6.1)-(6.2) we have

$\displaystyle\begin{split}{\mathtt{T}}_{0}(h,r,n)\in\qquad&\\\
\mathcal{P}^{\star}(r,n)\backslash\Pi_{\mathcal{P}^{\star}(r,n)}\\{&(Q,A,P)\in\mathcal{P}^{\star}(r,n)\times
O(n,m)\times\mathcal{P}(r,m)|\\\ &\text{Span }(A)\in{\mathsf{\Lambda}}_{0}\ ,\
\ P(x)=Q(Ax)\ ,\ \ P(x)\in\Sigma(r,1,m)\\}\ .\end{split}$ (9.3)

Hence, if claim (9.2) is true, taking (9.3) into account one retraces exactly
the same steps used to demonstrate Theorem A, (see section 6), and the thesis
follows. As it has already been pointed out in subparagraph 3.2.4, one just
needs to be careful about the fact that the steepness coefficients, in this
case, do not admit a uniform lower bound, as they may tend to zero when a
subspace not belonging to ${\mathsf{\Lambda}}_{0}(h,m,n)$ and on which $h$ is
not steep is approached. Moreover, one cannot expect to have steepness on the
subspaces of ${\mathsf{\Lambda}}_{0}(h,m,n)$ for an open set of functions
around $h$.

For any choice of $\Gamma^{m}\in{\mathsf{\Lambda}}_{0}(h,m,n)$, by Theorem 5.1
and by (5.37), condition

$({\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m),{\mathtt{a}},{{\mathtt{J}}_{1,\gamma,{\mathtt{a}}}})\not\in\Upsilon^{1}(\widehat{Z}^{1}(r,1,m))$

is equivalent to asking that system

${\mathtt{Q}}_{i\alpha}({\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m),{\mathtt{a}},{{\mathtt{J}}_{1,\gamma,{\mathtt{a}}}})=0\qquad\forall\
i\in\\{1,\dots,m\\}\ ,\ \ \forall\alpha\in\\{0,1\\}\ ,$ (9.4)

has no solution. Then, by expressions (7.8)-(7.9), the system in (9.4) is not
satisfied if and only if

$\begin{cases}(u_{1},\dots,u_{m})\in{\mathtt{U}}(m,n)\quad,\qquad\text{ Span
}(u_{1},\dots,u_{m})=\Gamma^{m}\\\
(a_{21},\dots,a_{m1})\in\overline{B}^{m-1}(0,{\mathscr{K}}(r,n,m))\quad,\qquad
v:=u_{1}+a_{21}u_{2}+\dots+a_{m1}u_{m}\\\
h_{0}^{1}[v]=h_{0}^{1}[u_{2}]=\dots=h^{1}_{0}[u_{m}]\\\
h^{2}_{0}[v,v]=h^{2}_{0}[v,u_{2}]=\dots=h^{2}_{0}[v,u_{m}]=0\end{cases}$ (9.5)

has no solution.

By construction, the Hessian of the restriction $h|_{\Gamma^{m}}$ has no null
eigenvalues, so that system (9.5) admits no solution, and both claim (9.2) and
expression (9.3) hold true, as wished. This concludes the proof.

###  9.2 Proof of Theorem C2

####  9.2.1 Case of a subspace belonging to ${\mathsf{\Lambda}}_{1}(h,m,n)$

With the usual setting, let $m\in\\{2,\dots,n-1\\}$ be an integer, and
$\Gamma^{m}\in{\mathsf{\Lambda}}_{1}(h,m,n)\subset{\mathtt{G}}(m,n)$ be a
$m$-dimensional subspace spanned by vectors
$u_{1},\dots,u_{m}\in{\mathtt{U}}(m,n)$.

As the Hessian matrix of the restriction $h|_{\Gamma^{m}}$ has at most one
null eigenvalue, without any loss of generality one can choose $u_{1}$ to be
the eigenvector associated to the unique null eigenvalue, that is

$\begin{cases}h^{1}_{0}[u_{1}]=h^{1}_{0}[u_{2}]=\dots=h_{0}^{1}[u_{m}]=0\\\
h_{0}^{2}[u_{1},u_{1}]=h^{2}_{0}[u_{1},u_{2}]=h^{2}_{0}[u_{1},u_{m}]=0\\\
\text{Span}(u_{1},u_{2},\dots,u_{m})=\Gamma^{m}\end{cases}$ (9.6)

so that at the same time one must have

$\det\left(\begin{matrix}h^{2}_{0}[u_{2},u_{2}]&h^{2}_{0}[u_{2},u_{3}]&\dots&h^{2}_{0}[u_{2},u_{m}]\\\
h^{2}_{0}[u_{3},u_{2}]&h^{2}_{0}[u_{3},u_{3}]&\dots&h^{2}_{0}[u_{3},u_{m}]\\\
\dots\\\
h^{2}_{0}[u_{m},u_{2}]&h^{2}_{0}[u_{m},u_{3}]&\dots&h^{2}_{0}[u_{m},u_{m}]\\\
\end{matrix}\right)\neq 0\ .$ (9.7)

The expression of ${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)$ w.r.t. the
coordinates $x_{1},\dots,x_{m}$ associated to the vectors
$u_{1},u_{2},\dots,u_{m}$ reads

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)(x)=\sum_{\begin{subarray}{c}\mu\in{\mathbb{N}}^{m}\\\
1\leq|\mu|\leq
r\end{subarray}}\frac{1}{\mu!}h_{0}^{|\mu|}\Big{[}\stackrel{{\scriptstyle\mu_{1}}}{{\overbrace{u_{1}}}},\stackrel{{\scriptstyle\mu_{2}}}{{\overbrace{u_{2}}}},\dots,\stackrel{{\scriptstyle\mu_{m}}}{{\overbrace{u_{m}}}}\Big{]}\,x_{1}^{\mu_{1}}\dots
x_{m}^{\mu_{m}}\ .$ (9.8)

We now claim that

######  Lemma 9.1.

If $h$ is non-steep at the origin on $\Gamma^{m}$ at some given order
$s_{m}\geq 2$, then

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in
X_{1}^{1}(r,s_{m},m):=\widehat{\sigma}^{1}(r,s_{m},m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)$
(9.9)

and ${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)$ satisfies the $s_{m}$-vanishing
condition on $\Gamma^{m}$ on some curve $\gamma\in\widehat{\Theta}_{m}^{1}$
whose Taylor expansion at the origin has null linear terms.

###### Proof.

Looking at (9.8), it is easy to check that the coefficients of
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)$ associated to the family of indices
$\\{\varpi(b,c)\\}_{\begin{subarray}{c}b,c\in\\{2,\dots,m\\},b\leq
c\end{subarray}}$ introduced in (8.12) read

$p_{\varpi(b,c)}=\left({\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\right)_{\varpi(b,c)}=\frac{1}{1+\delta_{bc}}h^{2}_{0}[u_{b},u_{c}]\
,$ (9.10)

where $\delta_{bc}$ is the Kronecker delta.

By putting together expressions (9.7) - (9.10) with the definition of set
${\mathsf{S}}_{1}^{1}(r,m)$ in (8.18), one has that

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in{\mathsf{S}}_{1}^{1}(r,m)\ .$ (9.11)

Since we have assumed $h$ is non-steep at the origin on $\Gamma^{m}$ at some
order $s_{m}\geq 2$ then, by the discussions at section 6 and at paragraph
8.1, one has

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}(r,s_{m},m)$ (9.12)

which, together with (9.11), yields

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}(r,s_{m},m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)\
.$ (9.13)

We now claim that

######  Lemma 9.2.

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\not\in\widehat{\Sigma}^{i}(r,s_{m},m)\qquad\forall\
i\in\\{2,\dots,m\\}\ .$ (9.14)

###### Proof.

Suppose, by absurd, that
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{i}(r,s_{m},m)$ for
some $i\in\\{2,\dots,m\\}$. It is clear from Theorem 5.1 and (8.9) that, as
$s_{m}\geq 2$, one has
$\widehat{\sigma}^{i}(r,s_{m},m)\subset\widehat{\sigma}^{i}(r,1,m)$, and
therefore $\widehat{\Sigma}^{i}(r,s_{m},m)\subset\widehat{\Sigma}^{i}(r,1,m)$.
This fact and the initial hypothesis imply
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{i}(r,1,m)$, so that
by Theorem 8.1, one must have

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\sigma}^{i}(r,1,m)\ .$
(9.15)

Relation (9.15) implies that there must exist a curve
$\gamma\in\widehat{\Theta}_{m}^{i}$ with values in $\Gamma^{m}$ such that
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)$ satisfies the $1$-vanishing condition
on $\gamma$. As it was shown in the proof of Theorem C1 (see the discussion
around formula (9.4)) this is equivalent to asking that system

$\begin{cases}h^{1}_{0}[u_{1}]=h^{1}_{0}[u_{2}]=\dots=h^{1}_{0}[u_{i}]=\dots=h^{1}_{0}[u_{m}]=0\\\
h^{2}_{0}[u_{i},u_{1}]=h^{1}_{0}[u_{i},u_{2}]=\dots=h^{1}_{0}[u_{i},u_{i}]=\dots=h^{1}_{0}[u_{i},u_{m}]=0\end{cases}\qquad
i\neq 1$ (9.16)

admits a solution, which contradicts (9.7). ∎

Due to (9.13) and to Lemma 9.2, we then have that

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{1}(r,s_{m},m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)\
.$ (9.17)

Moreover, by Theorem 8.2, the set
$X_{1}^{1}(r,s,m):=\widehat{\sigma}^{1}(r,s,m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)$
defined in (8.44) is closed in ${\mathsf{S}}_{1}^{1}(r,m)$ for the topology
induced by $\mathcal{P}(r,m)$, whence one deduces that actually

$X_{1}^{1}(r,s_{m},m)=\text{closure}\left(\widehat{\sigma}^{1}(r,s_{m},m)\right)\bigcap{\mathsf{S}}_{1}^{1}(r,m)=\widehat{\Sigma}^{1}(r,s_{m},m)\bigcap{\mathsf{S}}_{1}^{1}(r,m)\
.$ (9.18)

Relations (9.17) and (9.18) together imply

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in X_{1}^{1}(r,s_{m},m)\ .$ (9.19)

Therefore, by (9.19) and by the definition of $X^{1}_{1}(r,s_{m},m)$ in
(8.44), there exists a curve $\gamma\in\widehat{\Theta}_{m}^{1}$, with image
in $\Gamma^{m}$, on which the Taylor polynomial of the restriction
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)$ satisfies the $s_{m}$-vanishing
condition. Namely, if
${\mathtt{a}}=(a_{21},\dots,a_{m1})\in\overline{B}^{m-1}(0,{\mathscr{K}}(r,n,m))$
indicates the linear coefficients of $\gamma$ and
${\widehat{\mathtt{J}}_{s_{m},\gamma,{\mathtt{a}}}}$ its $s_{m}$-truncation
written in the adapted coordinates, by (8.9) one must have

$({\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m),{\mathtt{a}},{\widehat{\mathtt{J}}_{s_{m},\gamma,{\mathtt{a}}}})\in\Upsilon^{1}(\widehat{Z}^{1}(r,s_{m},m))\
,$ (9.20)

that is, by (5.37),

$\widehat{\mathtt{Q}}_{\ell\alpha}({\mathtt{T}}_{0,{\mathtt{a}}}(h|_{\Gamma^{m}},r,m),{\mathtt{a}},{\widehat{\mathtt{J}}_{s_{m},\gamma,{\mathtt{a}}}})=0\
,\ \ \ell=1,\dots,m\ ,\ \ ~\alpha=0,\dots,s_{m}\ .$ (9.21)

In particular, as $s_{m}\geq 2$, due to Theorem 5.1 and to (7.5), the
equations in (9.21) for $\alpha=0,1$ read

$\begin{cases}h^{1}_{0}[v]=h^{1}_{0}[u_{1}]=\dots=h_{0}^{1}[u_{m}]=0\\\
h_{0}^{2}[v,v]=h^{2}_{0}[v,u_{2}]=h^{2}_{0}[v,u_{m}]=0\\\
\text{Span}(v,u_{2},\dots,u_{m})=\Gamma^{m}\end{cases}$ (9.22)

where

$v=u_{1}+\sum_{i=2}^{m}a_{i1}u_{i}$ (9.23)

is the anisotropic vector associated to the curve $\gamma$. Comparing (9.6) to
(9.22), as the Hessian of $h|_{\Gamma^{m}}$ has only one null eigenvalue we
see that the vectors $u_{1}$ and $v$ must be parallel, but then (9.23) yields

$v=u_{1}\quad,\qquad a_{21}=\dots=a_{m1}=0\ ,$ (9.24)

so that by the arguments of subsection (5.2.3) the coordinates adapted to the
curve $\gamma$ coincide with the original coordinates.

∎

We now recall that, due to Theorem 8.2, the form of the set

$Y^{1}_{1}(r,s_{m},m):=\Pi_{\widehat{W}^{1}(r,m)}\left({\mathscr{N}}\left(\\{\widehat{\mathtt{Q}}_{{\mathtt{i}}\alpha}\\}_{\begin{subarray}{c}{\mathtt{i}}\in\\{1,\dots,m\\}\\\
\alpha\in\\{0,\dots,s_{m}\\}\end{subarray}}\right)\bigcap\Upsilon^{1}({\mathsf{S}}_{1}^{1}(r,m)\times\widehat{\vartheta}^{1}_{m}(s))\right)$
(9.25)

satisfying202020The transformation ${\mathfrak{U}}^{1}$ was introduced in
Remark 5.7.

$X^{1}_{1}(r,s_{m},m)=\Pi_{\mathcal{P}(r,m)}\left(({\mathfrak{U}}^{1})^{-1}Y^{1}_{1}(r,s_{m},m)\right)$
(9.26)

can be explicitly computed - starting from the relations determining
$\Upsilon^{1}(\widehat{Z}^{1}(r,s_{m},m))$ \- by the means of an algorithm
involving only linear operations. By this fact, the form of the set

$\mathscr{Y}^{1}_{1}(r,s_{m},m):=Y^{1}_{1}(r,s_{m},m)\bigcap\\{({\mathtt{p}}_{\mu},{\mathtt{a}})\in\widehat{W}^{1}(r,m)|{\mathtt{a}}=0\\}$
(9.27)

can also be deduced explicitly starting from
$\Upsilon^{1}(\widehat{Z}^{1}(r,s_{m},m))$. Then, due to Lemma 9.1 and to
(9.26), one has that

$\overset{\textstyle\text{$h$ non-steep at $0$}\mathstrut}{\text{at order
$s_{m}$ on $\Gamma^{m}$}}\ \Longrightarrow\
{\mathfrak{U}}^{1}({\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m),0)=({\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m),0)\in\mathscr{Y}^{1}_{1}(r,s_{m},m)\
.$ (9.28)

Moreover, we observe the following facts:

  1. 1.

in section 7, the explicit expression of set
${\mathcal{Z}}^{r,s_{m},m}_{n}\subset\mathcal{P}^{\star}(r,n)\times\mathbb{R}^{(m-1)s_{m}}\times{\mathscr{V}}^{1}(m,n)$
introduced in Corollary B2 is obtained by injecting into the explicit
expression for

$\Upsilon^{1}(\widehat{Z}^{1}(r,s_{m},m)):={\mathscr{N}}\left\\{{\mathtt{Q}}_{i,\alpha}({\mathtt{T}}_{0,{\mathtt{a}}}({\mathtt{P}}_{\mathtt{a}},r,m),{\mathtt{a}},{\mathtt{J}_{s_{m},\gamma,{\mathtt{a}}}})\right\\}_{\begin{subarray}{c}i=1,\dots,m\\\
\alpha=0,\dots,s_{m}\end{subarray}}$ (9.29)

given in Theorem 5.1 the explicit form of the coefficients of
${\mathtt{T}}_{0,{\mathtt{a}}}(Q|_{\Gamma^{m}},r,m)$ in (7.1)-(7.3) (which we
have written again in(9.1)), with $Q$ any polynomial belonging to
$\mathcal{P}^{\star}(r,n)$, and by considering the vectors
$v,u_{2},\dots,u_{m}$ in (9.1) as free parameters of ${\mathscr{V}}^{1}(m,n)$.

  2. 2.

In the same way, for any $Q\in\mathcal{P}^{\star}(r,n)$, one can inject in the
expressions determining $\mathscr{Y}^{1}_{1}(r,s_{m},m)$ the explicit form of
the coefficients of ${\mathtt{T}}_{0,{\mathtt{a}}}(Q|_{\Gamma^{m}},r,m)$,
given in (9.1), with the vectors $v,u_{2},\dots,u_{m}$ considered as free
parameters of ${\mathscr{V}}^{1}(m,n)$. The resulting subset is indicated by

${\mathscr{A}}_{1}(r,s_{m},n,m)\subset\mathcal{P}^{\star}(r,n)\times\mathbb{R}^{m-1}\times{\mathscr{V}}^{1}(m,n)\
.$

By the arguments above, and by the fact that the form of
$\mathscr{Y}^{1}_{1}(r,s_{m},m)$ is obtained starting from the expression of
$\Upsilon^{1}(\widehat{Z}^{1}(r,s_{m},m))$ through linear algorithms, we have
that the explicit expression of ${\mathscr{A}}_{1}(r,s_{m},n,m)$ can be found
starting from the expressions determining ${\mathcal{Z}}^{r,s_{m},m}_{n}$ by
only performing linear operations.

Furthermore, the above arguments together with (9.28) and with formula (9.24)
yield that if system

$\displaystyle\begin{split}&\begin{cases}(u_{1},\dots,u_{m})\in{\mathtt{U}}(m,n)\quad,\qquad\text{
Span }(u_{1},u_{2},\dots,u_{m})=\Gamma^{m}\in{\mathsf{\Lambda}}_{1}(h,m,n)\\\
({\mathtt{T}}_{0}(h,r,n),0,u_{1},u_{2},\dots,u_{m})\in{\mathscr{A}}_{1}(r,s_{m},n,m)\end{cases}\end{split}$
(9.30)

has no solution, then $h$ is steep at the origin with index $\alpha_{m}\leq
2s_{m}-1$ on any subspace $\Gamma^{m}\in{\mathsf{\Lambda}}_{1}(h,m,n)$.

####  9.2.2 Case of a subspace belonging to ${\mathsf{\Lambda}}_{2}(h,m,n)$

As we did in paragraph 9.2.1, we choose a basis
$u_{1},\dots,u_{m}\in{\mathtt{U}}(m,n)$ spanning $\Gamma^{m}$ such that
$u_{1}$ coincides with a normalized eigenvector of the Hessian associated to
one of the null eigenvalues. Then, we have

######  Lemma 9.3.

If $h$ is non-steep at the origin on
$\Gamma^{m}\in{\mathsf{\Lambda}}_{2}(h,m,n)$ at a given order $s_{m}\geq 2$,
up to suitably changing the order of the vectors $u_{1},\dots,u_{m}$ one has

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{1}(r,s_{m},m)\bigcap{\mathsf{S}}_{2}^{1}(r,m)\
.$ (9.31)

###### Proof.

By (9.10), and by the fact that the hessian of $h|_{\Gamma^{m}}$ has two or
more null eigenvalues, we have that

$\det\left(\begin{matrix}h^{2}_{0}[u_{2},u_{2}]&h^{2}_{0}[u_{2},u_{3}]&\dots&h^{2}_{0}[u_{2},u_{m}]\\\
h^{2}_{0}[u_{3},u_{2}]&h^{2}_{0}[u_{3},u_{3}]&\dots&h^{2}_{0}[u_{3},u_{m}]\\\
\dots\\\
h^{2}_{0}[u_{m},u_{2}]&h^{2}_{0}[u_{m},u_{3}]&\dots&h^{2}_{0}[u_{m},u_{m}]\\\
\end{matrix}\right)=0\ ,$ (9.32)

hence ${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in{\mathsf{S}}_{2}^{1}(r,m)$
following definition (8.20).

Moreover, $h$ is non-steep at the origin on $\Gamma^{m}$ at a given order
$s_{m}\geq 2$ so that, by the discussions at section 6 and at paragraph 8.1,
one has

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}(r,s_{m},m)$ (9.33)

so that by the previous considerations we have

${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}(r,s_{m},m)\bigcap{\mathsf{S}}_{2}^{1}(r,m)\
.$ (9.34)

By the above expression and by (8.9), we have that there must exist
$i\in\\{1,\dots,m\\}$ such that
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{i}(r,s_{m},m)\bigcap{\mathsf{S}}_{2}^{1}(r,m)$.
As it was already discussed in the proof of Theorem C1, if
${\mathtt{T}}_{0}(h|_{\Gamma^{m}},r,m)\in\widehat{\Sigma}^{i}(r,s_{m},m)\bigcap{\mathsf{S}}_{2}^{1}(r,m)$
then the vector $u_{i}$ must satisfy (9.16). If $i=1$, there is nothing else
to prove. If $i\neq 1$, it suffices to interchange the vector $u_{1}$ with the
vector $u_{i}$. ∎

Now, if for any $Q\in\mathcal{P}^{\star}(r,n)$ we inject in the
expressions212121Contrary to the case studied in the previous paragraph, here
these expressions cannot be found easily, in general. determining the semi-
algebraic subset
${\mathfrak{U}}^{1}\left(\left(\widehat{\Sigma}^{1}(r,s_{m},m)\bigcap{\mathsf{S}}_{2}^{1}(r,m)\right)\times\widehat{\vartheta}^{1}_{m}(1)\right)\subset\widehat{W}^{1}(r,m)$
the explicit expression of the coefficients of the polynomial
${\mathtt{T}}_{0,{\mathtt{a}}}(Q|_{\Gamma^{m}},r,m)$, and we let the vectors
$v,u_{2},\dots,u_{m}$ appearing in (9.1) vary in ${\mathscr{V}}^{1}(m,n)$, we
obtain a set

${\mathscr{A}}_{2}(r,s_{m},n,m)\subset\mathcal{P}(r,n)\times\mathbb{R}^{m-1}\times{\mathscr{V}}^{1}(m,n)\
.$

Moreover, by Lemma 9.3, we have that if system

$\displaystyle\begin{split}&\begin{cases}(a_{21},\dots,a_{m1})\in\overline{B}^{m-1}({\mathscr{K}})\\\
(u_{1},\dots,u_{m})\in{\mathtt{U}}(m,n)\quad,\qquad
v:=u_{1}+\sum_{i=2}^{m}a_{i1}u_{i}\\\ \text{ Span
}(v,u_{2},\dots,u_{m})=\Gamma^{m}\in{\mathsf{\Lambda}}_{2}(h,m,n)\\\
({\mathtt{T}}_{0}(h,r,n),a_{21},\dots,a_{m1},v,u_{2},\dots,u_{m})\in{\mathscr{A}}_{2}(r,s_{m},n,m)\end{cases}\end{split}$
(9.35)

has no solution, then $h$ is steep at the origin with index $\alpha_{m}\leq
2s_{m}-1$ on any subspace $\Gamma^{m}\in{\mathsf{\Lambda}}_{2}(h,m,n)$.

###  9.3 Proof of Theorem C3

####  9.3.1 Construction of an atlas of analytic maps for the Grassmannian

It is well known that for any pair of positive integers $k,n$, with $k<n$, the
Grassmannian ${\mathtt{G}}(k,n)$ has the structure of a projective algebraic
variety (see e.g. [19], [52]). In this subparagraph, for any positive integer
$n\geq 3$ and for any $m\in\\{2,\dots,n-1\\}$ we will construct a suitable
atlas of analytic maps for ${\mathtt{G}}(m,n)$ by exploiting classic arguments
of real-algebraic geometry.

Namely, we fix two integers $n\geq 3$, and $m\in\\{2,\dots,n-1\\}$ and we
consider a subset $I:=(i_{1},\dots,i_{m})\subset\\{1,\dots,n\\}$ of
cardinality $m$, as well as its complementary $J:=(j_{1},\dots,j_{n-m})$ of
cardinality $n-m$ in $\\{1,\dots,n\\}$.

We work in the euclidean space $\mathbb{R}^{n}$, and we fix a bilinear,
symmetric, non-degenerate form
$\mathsf{B}:\mathbb{R}^{n}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}$. The
Spectral Theorem ensures the existence of an orthonormal basis - indicated by
$e_{1},\dots,e_{n}$ \- that diagonalizes $\mathsf{B}$. Namely, in the basis
$e_{1},\dots,e_{n}$ the form $\mathsf{B}$ is represented by matrix

$\mathsf{B}_{(e_{1},\dots,e_{n})}:=\left(\begin{matrix}\alpha_{1}&0&0&\dots&0\\\
0&\alpha_{2}&0&\dots&0\\\ \dots&\dots&\dots&\dots&\dots\\\
0&0&0&\dots&\alpha_{n}\end{matrix}\right)\ ,$ (9.36)

where $\alpha_{1}\times\dots\times\alpha_{n}\neq 0$.

We indicate by $E_{I}$ (resp. $E_{J}$) the $m$-dimensional subspace spanned by
the vectors $(e_{i_{1}},\dots,e_{i_{m}})$ (resp. the $n-m$-dimensional
subspace spanned by $e_{j_{1}},\dots,e_{j_{n-m}}$). One clearly has
$\mathbb{R}^{n}=E_{I}\oplus E_{J}$. We also denote by $U_{J}$ the subset of
${\mathtt{G}}(m,n)$ containing the $m$-dimensional subspaces which are
supplementary for $E_{J}$.

With this setting, we consider the cartesian product
$E_{J}^{m}:=\stackrel{{\scriptstyle\text{$m$
times}}}{{\overbrace{E_{J}\times\dots\times E_{J}}}}$ and we have that

######  Lemma 9.4.

The map

$\mathscr{F}_{J}:E_{J}^{m}\longrightarrow
U_{J}\qquad(w_{1},\dots,w_{m})\longmapsto\text{Span}(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})$
(9.37)

is bijective.

###### Proof.

We proceed by steps. In the first two steps, we check that definition (9.37)
is well-posed. In Steps 3 and 4 we prove injectivity and surjectivity.

Step 1. We claim that for any choice of $(w_{1},\dots,w_{m})\in E_{J}^{m}$,
the vectors $(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})$ are linearly
independent. Infact, if by absurd there exist $m$ vectors
$(w_{1},\dots,w_{m})\in E_{J}^{m}$ such that
$(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})$ are not linearly independent, then
there must be a vector
$\lambda=(\lambda_{1},\dots,\lambda_{m})\in\mathbb{R}^{m}\backslash\\{0\\}$
satisfying $\sum_{\ell=1}^{m}\lambda_{\ell}(e_{i_{\ell}}+w_{\ell})=0$, that is
$\sum_{\ell=1}^{m}\lambda_{\ell}e_{i_{\ell}}=-\sum_{\ell=1}^{m}\lambda_{\ell}w_{i_{\ell}}$.

As $\sum_{\ell=1}^{m}\lambda_{\ell}e_{i_{\ell}}\in E_{I}$, and
$-\sum_{\ell=1}^{m}\lambda_{\ell}w_{i_{\ell}}\in E_{J}$, and as
$\mathbb{R}^{n}=E_{I}\oplus E_{J}$ by construction, by the assumptions one
must have
$-\sum_{\ell=1}^{m}\lambda_{\ell}w_{i_{\ell}}=\sum_{\ell=1}^{m}\lambda_{\ell}e_{i_{\ell}}=0$.
The previous relation - together with the fact that the vectors
$e_{i_{1}},\dots,e_{i_{m}}$ are linearly independent by hypothesis, yields
$\lambda=0$, in contradiction with the initial assumption $\lambda\neq 0$.

Step 2.  We claim that for any choice of $(w_{1},\dots,w_{m})\in E^{m}_{J}$
one has $\text{ Span }(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})\in U_{J}$.

By absurd, we suppose that for some $(w_{1},\dots,w_{m})\in E^{m}_{J}$ there
exist two non-zero vectors $u\in E_{J}$,
$\sigma=(\sigma_{1},\dots,\sigma_{m})\in\mathbb{R}^{m}$, verifying
$u=\sum_{\ell=1}^{m}\sigma_{\ell}(e_{i_{\ell}}+w_{\ell})$, that is
$\sum_{\ell=1}^{m}\sigma_{\ell}e_{i_{\ell}}=u-\sum_{\ell=1}^{m}\sigma_{\ell}w_{\ell}$.
By construction, one has $\sum_{\ell=1}^{m}\sigma_{\ell}e_{i_{\ell}}\in
E_{I}$, and $u-\sum_{\ell=1}^{m}\sigma_{\ell}w_{\ell}\in E_{J}$. Hence, as
$\mathbb{R}^{n}=E_{I}\oplus E_{J}$, the previous formula yields
$u-\sum_{\ell=1}^{m}\sigma_{\ell}w_{\ell}=\sum_{\ell=1}^{m}\sigma_{\ell}e_{i_{\ell}}=0$,
which in turn implies $\sigma=0$, as the vectors $(e_{i_{1}},\dots,e_{i_{m}})$
are linearly independent.

Hence, one has $E_{J}\cap\text{ Span
}(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})=\\{0\\}$. Therefore, since $\dim
E_{J}=n-m$ and $\dim\left(\text{ Span
}(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})\right)=m$ (the vectors
$e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m}$ are linearly independent by Step 1),
the subspace given by $\text{ Span }(e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m})$
is supplementary to $E_{J}$ and thus belongs to $U_{J}$.

Step 3. We prove that $\mathscr{F}_{J}$ is injective. By absurd, we suppose
that there exists a subspace in $U_{J}$ which has two different pre-images.
Namely, we suppose that there exist vectors $(u_{1},\dots,u_{m})\in
E_{J}^{m}$, and $(w_{1},\dots,w_{m})\in E_{J}^{m}$, with
$(u_{1},\dots,u_{m})\neq(w_{1},\dots,w_{m})$, satisfying
$\mathscr{F}_{J}(u_{1},\dots,u_{m})=\mathscr{F}_{J}(w_{1},\dots,w_{m})$.
Hence, as $e_{i_{1}}+u_{1},\dots,e_{i_{m}}+u_{m}$ and
$e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m}$ generate the same subspace, for any
$\ell\in\\{1,\dots,m\\}$ there must exist
$\beta^{\ell}=(\beta_{1}^{\ell},\dots,\beta_{m}^{\ell})\in\mathbb{R}^{m}\backslash\\{0\\}$
such that

$e_{i_{\ell}}+w_{\ell}=\sum_{k=1}^{m}\beta^{\ell}_{k}(e_{i_{k}}+u_{k})\ ,$
(9.38)

that is

$e_{i_{\ell}}-\sum_{k=1}^{m}\beta^{\ell}_{k}e_{i_{k}}=\sum_{i=1}^{m}\beta^{\ell}_{i}u_{i}-w_{\ell}\
.$ (9.39)

By construction one has
$e_{i_{\ell}}-\sum_{k=1}^{m}\beta^{\ell}_{k}e_{i_{k}}\in E_{I}$ and
$\sum_{i=1}^{m}\beta^{\ell}_{i}u_{i}-w_{\ell}\in E_{J}$, so due to (9.39) and
to the direct sum $\mathbb{R}^{n}=E_{I}\oplus E_{J}$ one infers
$\sum_{i=1}^{m}\beta^{\ell}_{i}u_{i}-w_{\ell}=e_{i_{\ell}}-\sum_{k=1}^{m}\beta^{\ell}_{k}e_{i_{k}}=0$.
Since the vectors $e_{i_{1}},\dots,e_{i_{m}}$ are linearly independent, we
finally obtain

$\beta^{\ell}_{k}=\delta_{\ell k}\ .$ (9.40)

where $\delta_{\ell k}$ is the Kronecker symbol. Formulas (9.38) and (9.40)
together imply that

$e_{i_{\ell}}+w_{\ell}=e_{i\ell}+u_{\ell}\quad\Longleftrightarrow\quad
w_{\ell}=u_{\ell}\qquad\forall\ell\in\\{1,\dots,m\\}\ ,$ (9.41)

in contradiction with the hypothesis
$(u_{1},\dots,u_{m})\neq(w_{1},\dots,w_{m})$.

Step 4.  We prove that $\mathscr{F}_{J}$ is surjective. Consider a subspace
$V\in U_{J}$. Since $V$ is supplementary of $E_{J}$, one has the direct sum
$\mathbb{R}^{n}=V\oplus E_{J}$, and for any $\ell\in\\{1,\dots,m\\}$ there
exist unique vectors $(v_{\ell},w_{\ell})\in V\times E_{J}$ such that
$e_{i_{\ell}}=v_{\ell}-w_{\ell}$. Hence, for any $\ell\in\\{1,\dots,m\\}$
there exists a unique choice of vectors $(w_{1},\dots,w_{m})$ satisfying
$v_{\ell}=e_{i_{\ell}}+w_{\ell}$. The vectors $v_{1},\dots,v_{m}$ belong to
$V$ by construction, and are linearly independent by Step 1. Therefore, since
$\dim V=m$ by hypothesis ($V\in U_{J}$), one has
$\text{Span}(v_{1},\dots,v_{m})=V$. ∎

We indicate by $\mathscr{J}^{n-m}$ the subsets of cardinality $n-m$ in
$\\{1,\dots,n\\}$. One has the following covering of the $m$-dimensional
Grassmannian:

${\mathtt{G}}(m,n)=\bigcup_{\begin{subarray}{c}J\in\mathscr{J}^{n-m}\end{subarray}}U_{J}\
.$ (9.42)

By construction, any vector $w_{\ell}\in E_{J}$ can be expressed uniquely as

$w_{\ell}=\sum_{k=1}^{n-m}M_{\ell k}e_{j_{k}}\ ,$ (9.43)

where $(M_{\ell k})_{\begin{subarray}{c}\ell=1,\dots,m\\\
k=1,\dots,n-m\end{subarray}}$ is a real $m\times(n-m)$ matrix. By (9.42), and
by Lemma 9.4, there exists an atlas sending ${\mathtt{G}}(m,n)$ to the open
union

$\bigcup_{\begin{subarray}{c}J\in\mathscr{J}^{n-m}\end{subarray}}\mathscr{F}_{J}^{-1}(U_{J})=\bigcup_{\begin{subarray}{c}J\in\mathscr{J}^{n-m}\end{subarray}}E_{J}^{m}\subset\mathbb{R}^{m\times(n-m)}\
.$ (9.44)

####  9.3.2 Proof of Theorem C3

Taking (9.42) into account, we fix $J\in\mathscr{J}^{n-m}$ together with its
associated sets $E_{J},U_{J}$. Let $V$ be a $m$-dimensional subspace belonging
to $U_{J}$. By Lemma 9.4, one has

$V=\text{ Span }\\{e_{i_{1}}+w_{1},\dots,e_{i_{m}}+w_{m}\\}$ (9.45)

for a unique $(w_{1},\dots,w_{m})\in E_{J}^{m}$.

Now, as ${\mathtt{G}}_{\geq 1}(m,n)$ contains those subsets of
${\mathtt{G}}(m,n)$ on which the restriction of the bilinear form $\mathsf{B}$
has at least one null eigenvalue, $V\in{\mathtt{G}}_{\geq 1}(m,n)$ if and only
if $\mathsf{B}$ is degenerated on $V$. Namely, taking (9.45) into account,
$V\in{\mathtt{G}}_{\geq 1}(m,n)$ iff there exists a vector
$v=\sum_{\ell=1}^{m}v_{\ell}(e_{i_{\ell}}+w_{\ell})\in V$ such that for all
$\ell^{\prime}\in\\{1,\dots,m\\}$ one has

$\displaystyle\begin{split}\mathsf{B}(v,e_{i_{\ell^{\prime}}}+w_{\ell^{\prime}})&=\sum_{\ell=1}^{m}v_{\ell}\,\mathsf{B}(e_{i_{\ell}}+w_{\ell},e_{i_{\ell^{\prime}}}+w_{\ell^{\prime}})\\\
&=\sum_{\ell=1}^{m}v_{\ell}\,\left(\mathsf{B}(e_{i_{\ell}},e_{i_{\ell^{\prime}}})+\mathsf{B}(w_{\ell},w_{\ell^{\prime}})\right)\\\
&=v_{\ell^{\prime}}\,\alpha_{i_{\ell^{\prime}}}+\sum_{\ell=1}^{m}v_{\ell}\,\mathsf{B}(w_{\ell},w_{\ell^{\prime}})\
.\end{split}$ (9.46)

To pass from the first to the second line in the above expression, we have
taken into account the fact that $(w_{1},\dots,w_{m})\in E_{J}^{m}$, that
$E_{J}=\text{Span}(e_{j_{1}},\dots,e_{j_{n-m}})$, and that the form
$\mathsf{B}$ is diagonal in the basis $e_{1},\dots,e_{m}$; in the last
passage, we have considered (9.36). Setting

$\mathscr{M}_{\mathsf{B}}=\mathscr{M}_{\mathsf{B}}(w_{1},\dots,w_{m}):=\left(\begin{matrix}\displaystyle\frac{1}{\alpha_{i_{1}}}\mathsf{B}(w_{1},w_{1})&\dots&\displaystyle\frac{1}{\alpha_{i_{1}}}\mathsf{B}(w_{1},w_{m})\\\
\dots&\dots&\dots\\\
\displaystyle\frac{1}{\alpha_{i_{m}}}\mathsf{B}(w_{m},w_{1})&\dots&\displaystyle\frac{1}{\alpha_{i_{m}}}\mathsf{B}(w_{m},w_{m})\\\
\end{matrix}\right)$ (9.47)

it is plain to check that (9.46) can be rewritten in the form

$\mathscr{M}_{\mathsf{B}}\,v=-v$ (9.48)

that is, we are asking for $-1$ to be an eigenvalue of
$\mathscr{M}_{\mathsf{B}}$, hence (9.48) is equivalent to

$\det(\mathscr{M}_{\mathsf{B}}+\mathbb{I}_{m})=0\ ,$ (9.49)

where $\mathbb{I}_{m}$ is the $m\times m$ identity matrix.

Since $\mathscr{M}_{\mathsf{B}}$ depends on $(w_{1},\dots,w_{m})\in E_{J}^{m}$
and since $(w_{1},\dots,w_{m})$ are in bijection with $\mathbb{R}^{m(n-m)}$ by
(9.43), the quantity $\det(\mathscr{M}_{\mathsf{B}}+\mathbb{I}_{m})$
determines a polynomial map $\mathbb{R}^{m(n-m)}\longrightarrow\mathbb{R}$.

If, by absurd, $\det(\mathscr{M}_{\mathsf{B}}+\mathbb{I}_{m})$ is the null
polynomial, then relation (9.49) holds on the whole inverse image
$\mathscr{F}_{J}^{-1}(U_{J})$. In particular, we observe that
$(w_{1}=0,\dots,w_{m}=0)\in\mathscr{F}_{J}^{-1}(U_{J})$ because
$\mathscr{F}_{J}(0,\dots,0)=\text{Span}\\{e_{i_{1}},\dots,e_{i_{m}}\\}=E_{I}$
and $E_{I}$ is supplementary of $E_{J}$ by construction. Therefore, by the
above reasonings one must have

$\det(\mathscr{M}_{\mathsf{B}}(0,\dots,0)+\mathbb{I}_{m})=\det\mathbb{I}_{m}=0$

which is clearly false. Consequently, the polynomial function
$\det(\mathscr{M}_{\mathsf{B}}(w_{1},\dots,w_{m})+\mathbb{I}_{m})$ is not
identically null over $\mathbb{R}^{m(n-m)}$ and, due to Lemma E.3, its zero
set is contained in a submanifold of codimension one in $\mathbb{R}^{m(n-m)}$.
Hence, also the subset of degenerated subspaces of $U_{J}$ is contained in a
submanifold of codimension one in ${\mathtt{G}}(m,n)$. The reasoning can be
repeated for all $J\in\mathscr{J}^{n-m}$. As, by its definition and by (9.42),
${\mathtt{G}}_{1}(m,n)$ is the finite union over $J\in\mathscr{J}^{n-m}$ of
the degenerated subspaces of $U_{J}$, we have that ${\mathtt{G}}_{1}(m,n)$ is
contained in a submanifold of codimension one in ${\mathtt{G}}(m,n)$. This
proves point 1 of the statement.

With the setting above, for any fixed $J\in\mathscr{J}^{n-m}$ we observe that
a $m$-dimensional subspace $V^{\prime}\in U_{J}$ belongs to
${\mathtt{G}}_{\geq 2}(m,n)$ if and only if there exist at least two linearly
independent vectors $v,u\in V^{\prime}$ satisfying (9.48). In particular, the
subset ${\mathtt{G}}_{\geq 2}(m,n)\cap U_{J}$ of "doubly-degenerated"
subspaces of $U_{J}$ is contained in the intersection of ${\mathtt{G}}_{\geq
1}(m,n)\cap U_{J}$ with the set

$T_{J}:=\\{W\in U_{J}|\Delta(P_{\mathscr{M}_{\mathsf{B}}})=0\\}$ (9.50)

where $P_{\mathscr{M}_{\mathsf{B}}}$ is the characteristic polynomial of
matrix $\mathscr{M}_{\mathsf{B}}$, and $\Delta(P_{\mathscr{M}_{\mathsf{B}}})$
is its discriminant. By the above arguments and by (9.42), point 2 of the
statement follows if we manage to prove that $T_{J}$ is contained in a
submanifold of codimension one in ${\mathtt{G}}(m,n)$. The rest of the proof
will be devoted to demonstrating this property.

Clearly, by the same arguments used in the proof of point 1 of the statement,
$\Delta(P_{\mathscr{M}_{\mathsf{B}}})$ is a polynomial function over
$\mathbb{R}^{m(n-m)}$. If, by absurd, $\Delta(P_{\mathscr{M}_{\mathsf{B}}})$
is identically zero in $\mathbb{R}^{m(n-m)}$, then in particular it must be
zero over the open set $\mathscr{F}^{-1}_{J}(U_{J})$.

Now, choose $m$ numbers $j_{1},\dots,j_{m}\in J$, and consider the vectors

$w^{\prime}_{1}:=\sqrt{\left|\frac{\alpha_{i_{1}}}{\alpha_{j_{1}}}\right|}e_{j_{1}},\quad
w^{\prime}_{2}:=\sqrt{2\left|\frac{\alpha_{i_{2}}}{\alpha_{j_{2}}}\right|}e_{j_{2}},\quad\dots,\quad
w^{\prime}_{m}:=\sqrt{m\left|\frac{\alpha_{i_{m}}}{\alpha_{j_{m}}}\right|}e_{j_{m}}\
,$ (9.51)

which are well defined by the fact that $\mathsf{B}$ is non-degenerate (see
(9.36)). It is clear that $(w^{\prime}_{1},\dots,w^{\prime}_{m})\in
E_{J}^{m}$, so that by Lemma (9.4) one has
$\mathscr{F}_{J}(w^{\prime}_{1},\dots,w^{\prime}_{m})\in U_{J}$. As
$\mathsf{B}$ is diagonal for the basis $e_{1},\dots,e_{n}$ by hypothesis,
matrix $\mathscr{M}_{\mathsf{B}}(w^{\prime}_{1},\dots,w^{\prime}_{m})$ in
(9.36) reads

$\left(\begin{matrix}\operatorname{sgn}(\alpha_{i_{1}})\operatorname{sgn}(\alpha_{j_{1}})&0&0&\dots&0\\\
0&2\operatorname{sgn}(\alpha_{i_{2}})\operatorname{sgn}(\alpha_{j_{2}})&0&\dots&0\\\
\dots&\dots&\dots&\dots&\dots\\\
0&0&0&\dots&m\operatorname{sgn}(\alpha_{i_{m}})\operatorname{sgn}(\alpha_{j_{m}})\\\
\end{matrix}\right)\ ,$ (9.52)

and it is clear that the discriminant of the characteristic polynomial of this
matrix cannot be zero, in contradiction with the fact that
$\Delta(P_{\mathscr{M}_{\mathsf{B}}})=0$ on the whole set
$\mathscr{F}_{J}^{-1}(U_{J})$. Therefore, the polynomial function
$\Delta(P_{\mathscr{M}_{\mathsf{B}}})$ is not identically null over
$\mathbb{R}^{m(n-m)}$ and its zero set is contained in a submanifold of
codimension one in $\mathbb{R}^{m(n-m)}$, by Lemma E.3. This proves that
$T_{J}$ is contained in a submanifold of codimension one in
${\mathtt{G}}(m,n)$, which concludes the proof.

## Tools of real-algebraic geometry

The goal of this appendix is to provide the reader with an overview of some
standard results of real-algebraic geometry that are used throughout the
present work. The interested reader can find a complete exposition in [19] and
[15].

##  Appendix A Semi-algebraic sets and semi-algebraic functions

######  Definition A.1.

A set $A\subset\mathbb{R}^{n}$ is said to be semi-algebraic if it can be
written in the form

$\bigcup_{i=1}^{s}\\{R_{i}(x)=0,Q_{i1}(x)<0,\dots,Q_{i\,r_{i}}(x)<0\\}\ ,$
(A.1)

where $R_{i},Q_{i1},...,Q_{i\,r_{i}}\in\mathbb{R}[x]$.

######  Remark A.1.

If only equalities are present in (A.1), $A$ is said to be algebraic.

It is clear that the polynomials generating a given semi-algebraic set $A$ are
not uniquely determined, nor is their number. However, one can introduce a
unique quantity associated to a semi-algebraic set, namely

######  Definition A.2 (Diagram).

For any semi-algebraic set $A\subset\mathbb{R}^{n}$, we denote by

  * $-$

$k_{1}(A)$ the minimal number of polynomials
$R_{i}(X),Q_{ij}(X)\in\mathbb{R}[X]$ that are necessary in order to determine
$A$ as in (A.1).

  * $-$

$k_{2}(A)$ the minimal value that the sum
$\sum_{i=1}^{s}\deg(\widehat{R}_{i})+\sum_{i=1}^{s}\sum_{j=1}^{r_{i}}\deg(\widehat{Q}_{ij})$
can attain, with $\widehat{R}_{i}(X),\widehat{Q}_{ij}(X)\in\mathbb{R}[X]$
determining $A$ as in (A.1).

Following [24] (Def. 9.1), we call diagram of $A$ the quantity

${\rm diag}(A):=k_{1}(A)+k_{2}(A)\ .$ (A.2)

Semi-algebraic sets are stable under projection, namely

######  Theorem A.1.

(Tarski and Seidenberg, quantitative version)

Take $n,m>0$ and let $A\subset\mathbb{R}^{n}\times\mathbb{R}^{m}$ be a semi-
algebraic set. We indicate by
$\Pi_{n}:\mathbb{R}^{n}\times\mathbb{R}^{m}\longrightarrow\mathbb{R}^{m},(x,y)\longmapsto
x$ the projector onto the first $n$ coordinates. Then, the set $\Pi_{n}(A)$ is
semi-algebraic and its diagram depends only on ${\rm diag}(A)$, $m$ and $n$.

The classic versions of the Theorem of Tarski and Seidenberg do not usually
make any reference to the diagram of the projected set. The statement given
here can be found in [24] (Proposition 9.2) and its proof is contained in
[14].

The Theorem of Tarski and Seidenberg is fundamental in order to demonstrate
the following results (see [19] for proofs)

######  Proposition A.1.

The complementary of a semi-algebraic set $A\subset\mathbb{R}^{n}$ is semi-
algebraic, and its diagram depends only on the diagram of $A$.

######  Proposition A.2.

The closure, the interior and the boundary of a semi-algebraic set
$A\subset\mathbb{R}^{n}$ are semi-algebraic and their diagrams depend only on
the diagram of $A$.

######  Proposition A.3.

Let $A$ be a semi-algebraic set of $\mathbb{R}^{n}$. Then, indicating with
$\overline{A}$ the closure of $A$, one has that ${\rm diag}(\overline{A})$
depends only on ${\rm diag}(A)$, and

$\dim A=\dim(\overline{A})\ .$

The notion of semi-algebraicness can be easily extended to functions by making
reference to their graphs, namely

######  Definition A.3.

Let $A\subset\mathbb{R}^{n}$ and $B\in\mathbb{R}^{m}$ be semi-algebraic sets.
A map $\varphi:A\longrightarrow B$ is said to be semi-algebraic if
$\text{graph}(\varphi)$ is a semi-algebraic set of
$\mathbb{R}^{n}\times\mathbb{R}^{m}$.

Semi-algebraic functions are piecewise algebraic, namely one has

######  Proposition A.4.

Let $A$ be a semialgebraic subset $A\subset\mathbb{R}^{n}$ and
$\varphi:A\longrightarrow\mathbb{R}$ be a semi-algebraic function of diagram
$d>0$. There exist a positive integer $M(d)$, a partition of $A$ into a finite
number of semi-algebraic sets $A_{i},i=1,\dots,m$, with $m\leq M(d)$, and for
every value of $i$ there exists a polynomial $S_{i}(X,Y)$ in $n+1$ variables
such that, for every $x$ in $A_{i}$, $S_{i}(x,Y)$ is not identically zero and
solves $S_{i}(x,\varphi(x))=0$.

###### Proof.

Except for the existence of the bound $M=M(d)$, the proof can be found in ref.
[19] (Lemma 2.6.3). Assume, by absurd, that the bound $M(d)$ does not exist;
then, one can find a sequence $\\{\varphi_{j}\\}_{j\in{\mathbb{N}}}$ of semi-
algebraic functions of $A$ with diagram $d$ such that, for each fixed
$j\in{\mathbb{N}}$, the Proposition holds with the minimal number of pieces in
the partition of $A$ being equal to $m_{j}$, and
$m_{j}\longrightarrow+\infty$. In particular, one can write
$A=\sqcup_{i=1}^{m_{j}}A_{i,j}$ for any $j\in{\mathbb{N}}$, and there exist
polynomials $S_{i,j}(X_{1},\dots,X_{n},Y)$ with the required properties. On
the one hand, for any given $j\in{\mathbb{N}}$ and $i\in\\{1,\dots,m_{j}\\}$,
by decomposition (A.1), one has that $\text{ graph}(\varphi_{j}|_{A_{i,j}})$
is the finite union of sets of the kind

$\displaystyle\begin{split}&\left\\{(x,y)\in A_{i,j}\times\mathbb{R}\ |\
S_{i,j}(x,y)=0,Q_{i1,j}(x,y)<0,\dots,Q_{ir_{i},j}(x,y)<0\right\\}\end{split}$
(A.3)

for some polynomials
$Q_{i1,j},\dots,Q_{ir_{i},j}\in\mathbb{R}[X_{1},\dots,X_{n},Y]$. On the other
hand, one has the disjoint union

$\text{graph}(\varphi_{j})=\bigsqcup_{i=1}^{m_{j}}\text{
graph}(\varphi_{j}|_{A_{i},j})\ ,$ (A.4)

which is a consequence of the fact that $A=\sqcup_{i=1}^{m_{j}}A_{i,j}$ is a
partition.

By Def. A.2, formulas (A.3)-(A.4), and the fact that $m_{j}$ is the minimal
number of pieces in the partition of $A$ for $\varphi_{j}$, one has ${\rm
diag}(\varphi_{j})\geq m_{j}$ and, for sufficiently high $j$, one has also
$m_{j}>d$ since $m_{j}\longrightarrow+\infty$, in contradiction with the
hypothesis ${\rm diag}(\varphi_{j})=d$ for any $j\in{\mathbb{N}}$. This
concludes the proof.

∎

An immediate consequence of Proposition A.4 is the following

######  Corollary A.1.

Let $A\subset\mathbb{R}$ be an interval (finite or infinite) and
$\varphi:A\longrightarrow\mathbb{R}$ be a semi-algebraic function of diagram
$d>0$. There exist a positive integer $M(d)$ and an interval ${\mathcal{I}}$
of length $|A|/M(d)$ over which the function $\varphi$ is algebraic, namely
there exists a polynomial $S(X,Y)$ in $n+1$ variables such that, for every $x$
in ${\mathcal{I}}$, $S(x,Y)$ is not identically zero and solves
$S(x,\varphi(x))=0$.

Among semi-algebraic functions, an important class is that of Nash functions:

######  Definition A.4.

Let $A$ be an open semi-algebraic subset of $\mathbb{R}^{n}$. A semi-algebraic
function $\varphi:A\longrightarrow\mathbb{R}$ belonging to the $C^{\infty}$
class is said to be a Nash function. The set of Nash functions on $A$ is a
ring under the usual operations of sum and function multiplication.

Moreover, if we define analytic-algebraic functions as those real-analytic
functions $f$ defined on an open semi-algebraic set $A\subset\mathbb{R}^{n}$
and satisfying $P(x,f(x))=0$ for some polynomial $P$ of $n+1$ variables and
for all $x\in A$, it turns out that

######  Proposition A.5 (ref. [19], Prop. 8.1.8).

A function $\varphi:A\longrightarrow\mathbb{R}$ is Nash on $A$ if and only if
it is analytic-algebraic.

Another important property of more general complex analytic-algebraic
functions is stated in the following

######  Proposition A.6.

Let $D\subset\mathbb{C}$ be an open, bounded domain. An analytic-algebraic
function $f:D\longrightarrow\mathbb{C}$, whose graph solves a polynomial
$S\in\mathbb{C}[z,w]$ of degree $k\in{\mathbb{N}}$, is $k$-valent: that is, if
$f$ is not constant then each value of ${\rm Im}(f)$ is the image of at most
$k$ points in $D$.

###### Proof.

Assume, by absurd, that $f$ is non-constant and that there exists
$w_{0}\in{\rm Im}(f)$ which is the image of at least $p>k$ points in $D$. The
polynomial $S^{w_{0}}(z):=S(z,w_{0})$ would admit $p>k$ roots while
${\deg}(S^{w_{0}})\leq k$ by hypothesis. The Fundamental Theorem of Algebra
ensures that $S^{w_{0}}$ must be identically zero and one has the
factorization $S(z,w)=(w-w_{0})^{\alpha}{\widehat{S}}(z,w)$, where
$\alpha\in\\{1,...,k\\}$, while $\widehat{S}$ cannot be divided by $(w-w_{0})$
in $\mathbb{C}[z,w]$. Since $f$ is analytic and not constant, then the
preimage $f^{-1}(\\{w_{0}\\})$ is a finite set and the graph of $f$ must
fulfill ${\widehat{S}}(z,f(z))=0$ out of $f^{-1}(\\{w_{0}\\})$. By continuity,
one has $\widehat{S}(z,f(z))=0$ on the whole domain of definition of $f$ since
$f^{-1}(\\{w_{0}\\})$ is finite. But $\deg\widehat{S}^{w_{0}}\leq k$, with
$\widehat{S}^{w_{0}}(z):=\widehat{S}(z,w_{0})$, and $\widehat{S}^{w_{0}}$
admits more than $k$ roots, hence the previous argument ensures that
$\widehat{S}$ can be divided by $(w-w_{0})$, in contradiction with the
construction. ∎

Semi-algebraicness is preserved by composition and inversion of semi-algebraic
maps. In the following propositions, $A\subset\mathbb{R}^{n}$ and
$B\subset\mathbb{R}^{m}$ are supposed to be semi-algebraic sets.

######  Proposition A.7.

Let $\varphi:A\longrightarrow B$ a semi-algebraic map. If $S\subset A$ and
$T\subset\varphi(A)$ are semi-algebraic, so are $\varphi(S)$ and the inverse
image $\varphi^{-1}(T)$. Moreover, their diagrams depend only on the diagram
of $\text{graph}(\varphi)$.

######  Proposition A.8.

Let $f:A\longrightarrow B$ and $g:B\longrightarrow C$ be two semi-algebraic
functions. Then $f\circ g$ is semi-algebraic and the diagram of its graph
depends only on the diagram of $\text{graph}(f)$ and on the diagram of
$\text{graph}(g)$.

######  Proposition A.9.

Let $f:A\longrightarrow B$ be an injective semi-algebraic function. Then, its
inverse $f^{-1}:f(A)\longrightarrow A$ is semi-algebraic and the diagram of
its graph depends only on the diagram of $\text{graph}(f)$.

######  Proposition A.10.

Let $I\subset\mathbb{R}$ be an open interval and
$f:I\longrightarrow\mathbb{R}$ be a semi-algebraic function differentiable in
$I$. Then its derivative $f^{\prime}$ is a semi-algebraic function and its
diagram only depends on the diagram of $\text{graph}(f)$.

We refer to [19] for the proofs of these statements. The dependence of the
diagrams on the diagram of the initial function is, once again, a consequence,
of the quantitative version A.1 of the Theorem of Tarski and Seidenberg.

Finally, we give the following statement, which will prove to be helpful in
our work

######  Proposition A.11.

(see e.g. ref. [39], pag. 23-24) Let $f:A\longrightarrow\mathbb{R}$ and
$g:A\longrightarrow\mathbb{R}^{m}$ be semi-algebraic functions and suppose
that $f$ is bounded from below. Then

$\varphi:g(A)\longrightarrow\mathbb{R}\qquad y\longmapsto\inf_{x\in
g^{-1}(y)}f(x)$

is semi-algebraic and the diagram of its graph depends only on the diagrams of
$\text{graph}(f)$ and $\text{graph}(g)$.

##  Appendix B Analytic reparametrization of semi-algebraic sets

Generally speaking, the reparametrization of a semi-algebraic set $A$ is a
subdivision of $A$ into semi-algebraic pieces $A_{j}$ each of which is the
image of a semi-algebraic function of the unit cube. On the one hand, it is
possible to cover the whole of $A$ if one asks for the covering functions to
be of finite regularity, with a uniform control on their derivatives (see
[35]). On the other hand, if one requires analyticity of the covering
functions together with a uniform control on their derivatives, it is only
possible to cover $A$ up to a "small" subset.

Hereafter, we state this result only in the case we need, that is for
reparametrizations of graphs of algebraic functions, referring to [59] for the
general theory.

It is known that algebraic functions can only have two type of complex
singularities: ramification points and poles (where the function may also
ramify). If we denote by $d$ the diagram of an algebraic function, the number
of its complex singularities is bounded by a quantity depending only on $d$
(see e.g. [13]). It is exactly the neighborhoods of these singularities that
cannot be analytically covered.

######  Definition B.1.

Let $\delta>0$ and $g:I:=[-1,1]\longrightarrow\mathbb{R}$ be an algebraic
function. An analytic $\delta$-reparametrization of $g$ consists of

  * $-$

A finite number $N$ of open subintervals $U_{i}$ of $I$, with
$\text{length}(U_{i})\leq 2\delta$ for any $i=1,...,N$.

  * $-$

A partition of $I\backslash\cup_{i=1}^{N}U_{i}$ into a finite number of
subsegments $\Delta_{j}$, $j=1,...,M$, together with a collection of real-
analytic maps $\psi_{j}:I\longrightarrow\Delta_{j}$ such that for any
$j\in\\{1,...,N\\}$

    1. 1.

$\psi_{j}$ is an affine reparametrization of the segment $\Delta_{j}$.

    2. 2.

$\psi_{j}$ and $g\circ\psi_{j}$ are both holomorphic in
${\mathcal{D}}_{3}:=\\{z\in\mathbb{C}:|z|\leq 3\\}\ .$

    3. 3.

$\psi_{j}$ and $g\circ\psi_{j}$ both satisfy a Bernstein inequality, that is

$\max_{z\in{\mathcal{D}}_{3}}|\psi_{j}(z)-\psi_{j}(0)|\leq
1\qquad\max_{z\in{\mathcal{D}}_{3}}|g\circ\psi_{j}(z)-g\circ\psi_{j}(0)|\leq
1\ .$ (B.1)

######  Theorem B.1.

(Yomdin, [59] Th. 3.2) Let $d$ be a positive integer. There exist constants
${\mathtt{Y}}_{1}={\mathtt{Y}}_{1}(d)$ and
${\mathtt{Y}}_{2}={\mathtt{Y}}_{2}(d)$ such that for each algebraic function
$g(x)$ of diagram $d$ defined on $I$ satisfying $0\leq g(x)\leq 1$ and for
each $\delta>0$ there is an analytic $\delta$-reparametrization of $g$ with
the number $N$ of the removed intervals bounded by ${\mathtt{Y}}_{1}$ and the
number $M$ of the covering maps bounded by
${\mathtt{Y}}_{2}\log_{2}(1/\delta)$. Each of the removed intervals is
centered at the real part of a complex singularity of $g(x)$.

Moreover, one has the following auxiliary result concerning the distance of
each interval $\Delta_{j}$ to the singularities of the complex extension of
the function $g$

######  Proposition B.1.

(Yomdin, [59], Lemma 3.6) The center of any of the segments $\Delta_{j}$ in
the partition given by Theorem B.1 is at distance no less than
$3\times|\Delta_{j}|$ from any complex singularity of $g$.

## Other auxiliary Results

##  Appendix C Quantitative local inversion theorem

We start by stating a Lipschitz inverse function Theorem. Its proof can be
found, for example, in [34] (Th. 14.6.6).

######  Theorem C.1.

Let $U$ be an open subset of a Banach space $E$ and that $k:U\longrightarrow
E$ is a Lipschitz mapping with constant $K<1$. Set $h(x)=x+k(x)$. If the
closed ball $\overline{\mathtt{B}}_{\varepsilon}(x)$ of radius $\varepsilon$
around $x$ is contained in $U$ then
$\overline{\mathtt{B}}_{(1-K)\varepsilon}(h(x))\subset
h(\overline{\mathtt{B}}_{\varepsilon}(x))\subset{\mathtt{B}}_{(1+K)\varepsilon}(h(x))$.
The mapping $h$ is a homeomorphism of $U$ onto $h(U)$, $h^{-1}$ is a Lipschitz
mapping with constant $1/(1-K)$, and $h(U)$ is an open subset of $E$.

This result is crucial in order to prove an analytic inverse function theorem,
namely

######  Theorem C.2.

Take a function $f\in C^{\omega}(\overline{\mathcal{D}}_{R}{(0)})$ and a point
$z^{*}\in{\mathcal{D}}_{R/2}{(0)}$ satisfying $f^{\prime}(z^{*})\neq 0$. Then,
$f$ is invertible in the closed disk
$\overline{\mathcal{D}}_{R^{\prime}/16}(z^{*})$ and its inverse $f^{-1}$ is
analytic in
$\overline{\mathcal{D}}_{|f^{\prime}(z^{*})|R^{\prime}/8}(f(z^{*}))$, where

$R^{\prime}:=\frac{1}{2}\times\min\left\\{R,\displaystyle\frac{|f^{\prime}(z^{*})|}{\max_{\overline{\mathcal{D}}_{R}(0)}|f^{\prime\prime}|},\right\\}\
.$

###### Proof.

We define $h(z):=\displaystyle\frac{f(z)-f(z^{*})}{f^{\prime}(z^{*})}+z^{*}$,
$k(z):=h(z)-z$; both these functions are obviously holomorphic in
$\overline{\mathcal{D}}_{R/2}(z^{*})$. Since
$k^{\prime}(z)=h^{\prime}(z)-1=\displaystyle\frac{f^{\prime}(z)}{f^{\prime}(z^{*})}-1=\frac{f^{\prime}(z)-f^{\prime}(z^{*})}{f^{\prime}(z^{*})}$,
one has
$|k^{\prime}(z)|=\displaystyle\frac{|f^{\prime}(z)-f^{\prime}(z^{*})|}{|f^{\prime}(z^{*})|}\leq\max_{\overline{\mathcal{D}}_{R}(0)}|f^{\prime\prime}|\
\frac{|z-z^{*}|}{|f^{\prime}(z^{*})|}$. If we choose to consider only the
$z\in{\mathcal{D}}_{R^{\prime}}(z^{*})$, we obtain that $k$ is
$\displaystyle\frac{1}{2}-$Lipschitz on this set.

At this point, we exploit Theorem C.1 and we have that the function
$h(z)=k(z)+z$ is a homeomorphism of ${\mathcal{D}}_{R^{\prime}}(z^{*})$ onto
its image. Moreover, one has
$\overline{{\mathcal{D}}}_{R^{\prime}/8}(h(z^{*}))\subset
h(\overline{{\mathcal{D}}}_{R^{\prime}/4}(z^{*}))\subset\overline{{\mathcal{D}}}_{3R^{\prime}/8}(h(z^{*}))$
and, since $h(z^{*})=z^{*}$, this yields

$\overline{{\mathcal{D}}}_{R^{\prime}/8}(z^{*})\subset
h(\overline{{\mathcal{D}}}_{R^{\prime}/4}(z^{*}))\subset\overline{{\mathcal{D}}}_{3R^{\prime}/8}(z^{*})\
.$ (C.1)

We can define $f^{-1}$ by exploiting $h$ and its inverse, namely

$z=h^{-1}(h(z))=h^{-1}\left(z^{*}+\displaystyle\frac{f(z)-f(z^{*})}{f^{\prime}(z^{*})}\right)=:f^{-1}(f(z))\
$ (C.2)

Indeed, by expressions (C.1) and (C.2), we see that, if we choose

$\left|z^{*}+\displaystyle\frac{f(z)-f(z^{*})}{f^{\prime}(z^{*})}-z^{*}\right|\leq\displaystyle\frac{R^{\prime}}{8}\
,$

that is
$\left|f(z)-f(z^{*})\right|\leq|f^{\prime}(z^{*})|\displaystyle\frac{R^{\prime}}{8}$,
we have defined the inverse over the closed disc
$\overline{\mathcal{D}}_{|f^{\prime}(z^{*})|R^{\prime}/8}(f(z^{*}))$. Finally,
we prove that

$f(\overline{\mathcal{D}}_{R^{\prime}/16}(z^{*}))\subset\overline{\mathcal{D}}_{|f^{\prime}(z^{*})|R^{\prime}/8}(f(z^{*}))\
.$

In order to see this, for $z\in\overline{\mathcal{D}}_{R^{\prime}/16}(z^{*})$
we consider the identity

$f(z)-f(z^{*})=f^{\prime}(z^{*})(z-z^{*})+\int_{0}^{1}f^{\prime\prime}(tz+(1-t)z^{*})\,(z-z^{*})^{2}\,(1-t)dt$

that yields the estimate

$\displaystyle\begin{split}|f(z)-f(z^{*})|\leq&|f^{\prime}(z^{*})||z-z^{*}|+\max_{\overline{\mathcal{D}}_{R}(0)}|f^{\prime\prime}|\,|z-z^{*}|^{2}\\\
\leq&|f^{\prime}(z^{*})|\frac{R^{\prime}}{16}+\max_{\overline{\mathcal{D}}_{R}(0)}|f^{\prime\prime}|\frac{R^{\prime
2}}{256}\leq|f^{\prime}(z^{*})|\frac{R^{\prime}}{8}\ ,\end{split}$ (C.3)

where the last estimate is a consequence of the definition of $R^{\prime}$.

The fact that $f^{-1}$ inherits the same regularity of $f$ is a standard
consequence of the classic local inversion theorem. ∎

##  Appendix D Two elementary properties of Lie Groups

We refer to [43] (Cor. 21.6, Th. 21.10) for proofs.

######  Proposition D.1.

Every continuous action by a compact Lie group on a manifold is proper.

######  Theorem D.1 (Quotient manifold).

Suppose $G$ is a Lie group acting smoothly, freely, and properly on a smooth
manifold ${\mathcal{M}}$. Then the orbit space ${\mathcal{M}}/G$ is a
topological manifold of dimension equal to $\dim{\mathcal{M}}-\dim G$ and has
a unique smooth structure with the property that the quotient map
$\pi:{\mathcal{M}}\longrightarrow{\mathcal{M}}/G$ is a smooth submersion.

##  Appendix E Three auxiliary Lemmas

The following Lemma is an elementary criterion to establish when the
projection of a closed set is still closed.

######  Lemma E.1.

Let $E$ be a metric space, $K$ a compact subset of some metric space and
$\Delta$ a closed subset of $E\times K$. Then, the projection of $\Delta$ on
$E$, indicated by $\Pi_{E}(\Delta)$, is closed.

###### Proof.

Let $\\{p_{n}\\}_{n\in\mathbb{N}}$ be a sequence in $\Pi_{E}(\Delta)$
converging to a point $\bar{p}$ and $\\{k_{n}\\}_{n\in\mathbb{N}}$ a sequence
in $K$ satisying $(p_{n},k_{n})\in\Delta$. Since $K$ is a compact subset of
some metric space, one can extract a subsequence
$\\{k_{n_{l}}\\}_{l\in\mathbb{N}}$ converging to a point $\bar{k}\in K$.
Hence, the sequence $\\{(p_{n_{l}},k_{n_{l}})\\}_{l\in\mathbb{N}}$ in $\Delta$
converges to $(\bar{p},\bar{k})\in\Delta$, since $\Delta$ is closed. This
implies that $\bar{p}$ belongs to $\Pi_{E}(\Delta)$, which is therefore
closed. ∎

The following statement is a known Theorem due to Bézout (see [42], Th. 3.4a).

######  Lemma E.2.

For any couple of positive integers $k_{1},k_{2}$ consider two non-zero
irreducible, non-proportional polynomials $Q_{1}\in\mathcal{P}(k_{1})$ and
$Q_{2}\in\mathcal{P}(k_{2})$. Then the system $Q_{1}(z,w)=Q_{2}(z,w)=0$ has at
most $k_{1}\times k_{2}$ solutions.

For the sake of completeness, we also state the following simple result on the
codimension of the zero set of a non-null polynomial.

######  Lemma E.3.

Let $n$ be a positive integer. Consider a non-null real polynomial
$P\in\mathbb{R}[x]$ of the variables $x=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$.
The zero set $Z_{P}:=\\{x\in\mathbb{R}^{n}|P(x)=0\\}$ is contained in a
submanifold of codimension one in $\mathbb{R}^{n}$.

###### Proof.

If $P$ is a non-zero constant, then there is nothing to prove.

If $P$ is non-constant, the proof is by induction on the degree of $P$.

If $\deg P=1$, then $Z_{P}$ is a hyperplane, which is obviously a submanifold
of codimension one.

Suppose, now, that the statement is true for polynomials of degree $k-1\geq
1$. Consider a polynomial of degree $k$, together with its associated open set
of non-critical points $S_{P}:=\\{x\in\mathbb{R}^{n}\,|\,\nabla P(x)\neq
0\\}$. On the one hand, locally around any point of $Z_{P}\cap S_{P}$ one can
apply the implicit function theorem, so that $Z_{P}\cap S_{P}$ is indeed a
submanifold of codimension one in $\mathbb{R}^{n}$. On the other hand,
$\mathbb{R}^{n}\backslash S_{P}$ is the common zero set of the $n$ polynomials
$\partial P/\partial x_{1},\dots,\partial P/\partial x_{n}$; moreover, since
$\deg P=k\geq 2$, at least one among $\partial P/\partial x_{1},\dots,\partial
P/\partial x_{n}$ has degree $k-1\geq 1$. Hence, by hypothesis,
$\mathbb{R}^{n}\backslash S_{P}$ is contained in a submanifold of codimension
one in $\mathbb{R}^{n}$.

This proves that the set $Z_{P}\cap(\mathbb{R}^{n}\backslash
S_{P})\subset(\mathbb{R}^{n}\backslash S_{P})$ is contained in a submanifold
of codimension one in $\mathbb{R}^{n}$. Obviously, the thesis follows by the
fact that

$Z_{P}=(Z_{P}\cap(\mathbb{R}^{n}\backslash S_{P}))\cup(Z_{P}\cap S_{P})\ .$

∎

## Acknowledgements

I am extremely grateful to my PhD advisors, Laurent Niederman and Luca Biasco,
for putting me on this subject, for their useful suggestions and comments, and
for reading and correcting the present manuscript. I am also grateful to J.P.
Marco for very useful discussions. In the months preceeding the end of the
redaction of this work, I have been funded by the ERC project 757802
Haminstab; therefore, I wish to acknowledge both the ERC and the PI of the
project (prof. M. Guardia) for their support.

## References

  * [1] V. Arnol’d.  The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case.  Soviet. Math. Dokl., 2:247–249, 1961.
  * [2] V. Arnol’d.  Instability of dynamical systems with several degrees of freedom.  Soviet. Math. Dokl., 156(1):9–12, 1964.
  * [3] V. Arnol’d, V. Kozlov, A.I. Neishtadt, and E. Khukhro.  Mathematical Aspects of Classical and Celestial Mechanics.  Springer, 2010.
  * [4] D. Bambusi.  Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations .  Math Z., 230:345–387, 1999.
  * [5] D. Bambusi and A. Fusè.  Nekhoroshev Theorem for perturbations of the central motion.  Regular and Chaotic Dynamics, 22:18–26, 2017.
  * [6] D. Bambusi, A. Fusè, and M. Sansottera.  Exponential stability in the perturbed central force problem.  Regular and Chaotic Dynamics, 23:821–841, 2018.
  * [7] D. Bambusi and A. Giorgilli.  Exponential stability of states close to resonance in infinite dimensional hamiltonian systems.  Journal of Statistical Physics, 71:569–606, 1993.
  * [8] D. Bambusi and B. Langella.  A $c^{\infty}$ nekhoroshev theorem.  Mathematics in Engineering, 3(2):1–17, 2020.
  * [9] D. Bambusi and B. Langella.  Growth of Sobolev Norms in quasi-integrable quantum systems.  https://arxiv.org/abs/2202.04505, 2022.
  * [10] S. Barbieri.  On the algebraic properties of exponentially stable integrable Hamiltonian systems.  Annales de la Faculté de Sciences de Toulouse, 31(5):1365–1390, 2022.
  * [11] S. Barbieri, J.P. Marco, and J.E. Massetti.  Analytic smoothing and Nekhoroshev estimates for Hölder steep Hamiltonians .  Commun. Math. Phys., 396(1), 2022.
  * [12] S. Barbieri and L. Niederman.  Sharp Nekhoroshev estimates for the three-body problem around periodic orbits.  Journal of Differential Equations, 268(7):3749–3780, 2020.
  * [13] S. Barbieri and L. Niederman.  Bernstein-Remez inequality for algebraic functions: a topological approach.  Nonlinear Anaysis, (237), 2023.
  * [14] S. Basu, R. Pollack, and M.-F. Roy.  On the combinatorial and algebraic complexity of quantifier elimination.  J. ACM, (45):1002–1045, 1996.
  * [15] S. Basu, R. Pollack, and M.-F. Roy.  Algorithms in real-algebraic geometry.  Springer, 2006.
  * [16] G. Benettin, F. Fassò, and M. Guzzo.  Nekhoroshev stability of L4 and L5 in the spatial restricted three body problem.  Regular Chaotic Dyn., (3):56–72, 1998.
  * [17] M. Berti, L. Biasco, and P. Bolle.  Drift in phase space: a new variational mechanism with optimal diffusion time.  Journal de mathématiques pures et appliquées, 82(6):613–664, 2003\.
  * [18] G. Binyamini and D. Novikov.  Complex cellular structures.  Ann. of Math., 190:145–248, 2019.
  * [19] J. Bochnak, M. Coste, and M.-F. Roy.  Real algebraic geometry.  Springer, 1998.
  * [20] J. Bolte, A. Daniilidis, O. Ley, and L. Mazet.  Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity.  Trans. Amer. Math. Soc., 362:3319–3363, 2010.
  * [21] A. Bounemoura.  Nekhoroshev estimates for finitely differentiable, quasi-convex Hamiltonian systems.  Journal of Differential Equations, 249(11):2905–2920, 2010.
  * [22] A. Bounemoura, B. Fayad, and L. Niederman.  Super-exponential stability for generic real-analytic elliptic equilibrium points.  Advances in Mathematics, 366(3), 2020.
  * [23] A. Bounemoura and V. Kaloshin.  Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom.  Mosc. Math. J., 14(2):181–203, 426, 2014.
  * [24] J. Bourgain.  Green’s functions estimates for lattice Schrödinger operators and applications.  Princeton University Press, 2005.
  * [25] J. Bourgain, M. Goldstein, and W. Schlag.  Anderson localization for Schrödinger operators on $\mathbb{Z}^{2}$ with quasi-periodic potential.  Acta Math., (188):41–86, 2002.
  * [26] H.W. Broer, G.B. Huitema, and M. Sevryuk.  Quasi-periodic motions in families of dynamical systems - Order amidst Chaos.  Springer, 1996.
  * [27] D. Burguet.  A proof of Yomdin-Gromov’s algebraic lemma .  Israel Journal of Mathematics, 168:291–316, 2008.
  * [28] L. Chierchia.  Kolmogorov’s 1954 paper on nearly-integrable Hamiltonian systems.  Regular and Chaotic Dynamics, 13(2):130–139, 2008.
  * [29] L. Chierchia, M. A. Faraggiana, and M. Guzzo.  On steepness of 3-jet non-degenerate functions.  Ann. Mat. Pura Appl. (4), (6):2151–2165, 2019.
  * [30] R. Cluckers, J. Pila, and A. Wilkie.  Uniform parameterization of subanalytic sets and diophantine applications.  Ann. Sci. Éc. Norm. Super., 53:1–42, 2020.
  * [31] E. Faou and B. Grébert.  A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus .  Analysis and Partial Differential Equations, 6(6):1243–1262, 2013\.
  * [32] B. Fayad.  Lyapunov unstable elliptic equilibria.  J. Amer. Math. Soc., 36(1):81–106, 2023.
  * [33] J. Féjoz.  Démonstration du Théorème d’Arnol’d sur la stabilité du système planétaire.  Ergodic Theory Dynam. Systems, 24(5):1521–1582, 2004.
  * [34] D.J.H. Garling.  A course in mathematical analysis.  Cambridge University Press, 2014.
  * [35] M. Gromov.  Entropy, homology and semi-algebraic geometry.  Astérisque - Séminaire Bourbaki 663, pages 225–240, 1987.
  * [36] M. Guzzo, L. Chierchia, and G. Benettin.  The Steep Nekhoroshev’s Theorem.  Commun. Math. Phys., 342:569–601, 2016.
  * [37] J. Heintz, M. Roy, and P. Solernó.  Sur la complexité du principe de Tarski-Seidenberg.  Bulletin de la S. M. F., 118(1):101–126, 1990.
  * [38] M.-R. Herman.  Dynamics connected with indefinite normal torsion.  In Twist mappings and their applications, volume 44 of IMA Vol. Math. Appl., pages 153–182. Springer, New York, 1992.
  * [39] H.-V. Hà and T. Pham.  Genericity in Polynomial Optimization.  World Scientific, 2017.
  * [40] Y. Ilyashenko.  A criterion of steepness for analytic functions.  Uspekhi Mat. Nauk., 41:193–194, 1986.
  * [41] V. Kaloshin and K. Zhang.  Arnold diffusion for smooth systems of two and a half degrees of freedom, volume 208 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2020.
  * [42] K. Kendig.  A guide to plane algebraic curves.  Mathematical Association of America, 2012.
  * [43] J.M. Lee.  Introduction to smooth manifolds.  Springer, 2013.
  * [44] F. Malige, P. Robutel, and J. Laskar.  Partial reduction in the $n$-body planetary problem using the angular momentum integral.  Celestial Mech. Dynam. Astronom., 84(3):283–316, 2002.
  * [45] J-P. Marco and D. Sauzin.  Stability and instability for gevrey quasi-convex near-integrable hamiltonian systems.  Publ. Math. Inst. Hautes Études Sci., 96:199–275, 2003.
  * [46] J. Moser.  On invariant curves of area-preserving mappings of an annulus.  Nachr. Akad. Wiss. Göttingen, II:1–20, 1962.
  * [47] N. N. Nekhoroshev.  Stable lower estimates for smooth mappings and for gradients of smooth functions.  Math USSR Sb., 19(3):425–467, 1973.
  * [48] N. N. Nekhoroshev.  An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. I.  Russian Mathematical Surveys, 32(6):1–65, 1977.
  * [49] N. N. Nekhoroshev.  An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. II.  Topics in Modern Mathematics, Petrovskii Seminar, (5):5–50, 1979\.
  * [50] L. Niederman.  Stability over exponentially long times in the planetary problem.  Nonlinearity, 9:1703–1751, 1996.
  * [51] L. Niederman.  Hamiltonian stability and subanalytic geometry .  Annales de l’Institut Fourier, 56(3):795–813, 2006.
  * [52] D. Perrin.  Algebraic Geometry. An Introduction. Springer, 2008.
  * [53] G. Pinzari.  Aspects of the planetary Birkhoff normal form.  Regular and Chaotic Dynamics, 18(6):860–906, 2013.
  * [54] J. Pöschel.  On Nekhoroshev Estimates for a Nonlinear Schrödinger Equation and a Theorem by Bambusi .  Nonlinearity, 12:1587–1600, 1999.
  * [55] N. Roytwarf and Y. Yomdin.  Bernstein Classes.  Annales de l’Institut Fourier, 47(3):825–858, 1998.
  * [56] H. Rüssmann.  Invariant tori in non-degenerate nearly integrable Hamiltonian systems.  Regular and Chaotic Dynamics, 6(2):119–204, 2001.
  * [57] G. Schirinzi and M. Guzzo.  On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev.  Journal of Mathematical Physics, 54(7):1–23, 2013.
  * [58] Y. Yomdin.  $C^{k}$-resolution of semialgebraic mappings. Addendum to: Volume growth and entropy.  Israel J. Math., 57:301–317, 1987.
  * [59] Y. Yomdin.  Analytic reparametrization of semi-algebraic sets.  Journal of Complexity, (24):54–76, 2008.
  * [60] Y. Yomdin.  Remez-type inequality for discrete sets.  Israel J. Math., 186:45–60, 2011.
  * [61] Y. Yomdin.  Smooth parametrizations in dynamics, analysis, diophantine and computational geometry.  Japan Journal Ind. Appl. Math., 32:411–435, 2015.