# Join operation and ${\mathcal{A}}$-finite map-germs

Rodrigues Hernandes, M. E. and Ruas, M. A. S The first author was partially
supported by FAPESP Proc. $2019/07316-0$ and the second author by FAPESP Proc.
$2019/21181-0$ and CNPq Proc. $305695/2019-3$.

###### Abstract

In this work we define some map-germs, called elementary joins, for the
purpose of producing new ${\mathcal{A}}$-finite map-germs from them. In
particular, we describe a general form of an ${\mathcal{A}}$-finite monomial
map from $(\mathbb{C}^{n},0)$ to $(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any
corank in terms of elementary join maps. Our main tools are the delta
invariant and some invariants of curves.

Keywords: _join operation, parameterized curves, analytic invariants._

##  1 Introduction

The class of map-germs from $(\mathbb{K}^{n},0)$ to $(\mathbb{K}^{2n},0)$,
where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$, has been investigated by
many authors using different approaches. For instance, Gaffney gives in [4] a
formula for the number of double points at the origin of a stable perturbation
of a map-germ $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{2n},0)$ in terms of the
zero dimensional Segre number of the double point ideal of $f$ and the number
of Whitney umbrella points of the composition of $f$ with a generic projection
from $\mathbb{C}^{2n}$ to $\mathbb{C}^{2n-1}$. Klotz, Pop and Rieger classify
${\mathcal{A}}$-simple germs $(\mathbb{R}^{n},0)\to(\mathbb{R}^{2n},0)$ (see
[8]) to show that all of them have a real deformation with the maximal number
of real double-points. Symplectic invariants of curves and surfaces in this
class of maps were also studied in [7] by Ishikawa and Janeczko. Parameterized
singular surfaces in $\mathbb{R}^{4}$ appear in [2], where Birbrair, Mendes
and Nuño-Ballesteros study relations between topological and metric
properties, and also in [1] where Benedini, Ruas and Sinha study their second
order geometry. In [13], the authors classify ${\mathcal{A}}$-finitely
determined monomial surfaces in $\mathbb{C}^{4}$ and so many others.

In this work we define an “operation”, called join operation, among a finite
number of map-germs in order to obtain map-germs that behave well with respect
to ${\mathcal{A}}$-equivalence. For two map-germs $F$ and $G$, this operation
consists in joining them by means of a germ $H$, denoted $(F*G)_{H}$, defined
as $(F*G)_{H}(x,y):=(F(x),G(y),H(x,y))$ which preserve in some sense the
properties of $F$ and $G$. Our main purpose is to define basic germs, called
elementary joins, in order to produce new ${\mathcal{A}}$-finitely determined
(or ${\mathcal{A}}$-finite) map-germs. Different type of operations to obtain
new ${\mathcal{A}}$-finite germs from $\mathbb{C}^{n}$ to $\mathbb{C}^{p}$
from germs in lower dimensions were defined in [3], see also [11].

The elementary join maps are obtained joining the identity and a parameterized
curve or a finite number of parameterized curves. In Proposition 3.2 and 3.4
we prove that these maps are ${\mathcal{A}}$-finite. In Theorem 4.2 and
Corollaries 4.3 and 4.6 we classify some classes of ${\mathcal{A}}$-finite
monomial map-germs. Moreover, we obtain a general form of an
${\mathcal{A}}$-finite monomial map-germ
$f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any corank in
terms of elementary joins. More precisely, in Theorem 4.4, we prove that if
$f$ is a corank $q$ monomial map-germ, then $f$ is ${\mathcal{A}}$-finite if
and only if $f$ is ${\mathcal{A}}$-equivalent to the join of the identity in
$\mathbb{C}^{n-q}$ with $q$ parameterized curves. These monomial normal forms
can be seen as ${\mathcal{A}}$-finite reflection maps, as defined in [12]. As
a corollary of Theorem 4.4 we obtain infinitely many families of
${\mathcal{A}}$-finite map-germs, not necessarily monomial. Two fundamental
tools to obtain the previous results were the invariants: the semigroup
associated to a parameterized curve and the delta invariant
$\delta_{f}:=dim_{\mathbb{C}}\frac{{\cal O}_{n}}{f^{*}({\cal O}_{p})}$,
defined in [13].

In Section 5, our focus is to study some ${\mathcal{A}}$-invariants of map-
germs $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ for $p\geq 2n$. The delta
invariant is also useful for verifying whether a map is
${\mathcal{A}}$-finitely determined, since $\delta_{f}$ is easier to be
calculated than the ${\mathcal{A}}_{e}$-codimension of $f$
(${\mathcal{A}}_{e}cod(f)$). If $p=2n$, we also have the number $d(f)$ of
double points appearing in a stable perturbation of $f$. The results of this
section provide estimates of $\delta_{f}$ and consequently for ${\cal
A}_{e}cod(f)$ and some formulas for $d(f)$. In Subsection 5.1, we consider $f$
${\mathcal{A}}$-equivalent to $(g_{k},h)$, where $g_{k}=\pi_{k}\circ f$ in
which $\pi_{k}:\mathbb{C}^{p}\to\mathbb{C}^{k}$ is a canonical projection and
$h:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p-k},0)$. Such decomposition can be
interesting in the study of analytic invariants, if one knows some properties
of $g_{k}$ and $h$. In Theorem 5.5, we prove that
${\mathcal{A}}_{e}cod(f)\leq{\mathcal{A}}_{e}cod(g_{k})+(p-k)\delta_{f}$, with
a formula when $g_{k}$ is a stable map. If the projection $g_{n}$ of $f$ is a
fold map, we prove in Proposition 5.6 that $d(f)$ and $\delta_{f}$ coincide
and we provide upper and lower bounds for ${\mathcal{A}}_{e}cod(f)$.

##  2 Basic notations of singularities

Analytical invariants play a central role in this work. The delta invariant of
a map-germ and some invariants of the theory of curves such as the semigroup
of values and its set of gaps are fundamental for the characterization of some
classes of map-germs.

Let ${\cal O}(n,p)$ be the set of smooth map-germs
$f:(\mathbb{C}^{n},0)\to\mathbb{C}^{p}$, which is a free ${\cal O}_{n}$-module
of rank $p$, where ${\cal O}_{n}$ is the local ring of function germs in
$\mathbb{C}^{n}$ at the origin, with maximal ideal ${\cal M}_{n}$. Denoting by
${\mathcal{R}}=Diff(\mathbb{C}^{n},0)$ and
${\mathcal{L}}=Diff(\mathbb{C}^{p},0)$ the groups of germs of diffeomorphisms
at the origin, we consider the set ${\cal M}_{n}{\cal O}(n,p)=\\{f\in{\cal
O}(n,p);f(0)=0\\}$ under the action of the group
${\mathcal{A}}={\mathcal{R}}\times{\mathcal{L}}$. Two map-germs $f,g\in{\cal
M}_{n}{\cal O}(n,p)$ are called ${\mathcal{A}}$-equivalent, denoted
$f\underset{{\mathcal{A}}}{\sim}g$, if there exist
$(\rho,\psi)\in{\mathcal{A}}$ such that $g=\psi\circ f\circ\rho^{-1}$.

Let $\theta(f)=\\{\sigma:(\mathbb{C}^{n},0)\to T\mathbb{C}^{p};\
\pi\circ\sigma=f\\}$ such that $\pi:T\mathbb{C}^{p}\to(\mathbb{C}^{p},0)$ is
the natural projection and $\theta(n)$ the set of germs of vector fields in
$\mathbb{C}^{n}$ at the origin. We define $tf:\theta(n)\to\theta(f)$ by
$tf(\epsilon)(x)=df_{x}(\epsilon(x))$, where $df$ is the differential of $f$
and $wf:\theta{(p)}\to\theta(f)$ such that $wf(\eta)=f^{*}(\eta):=\eta\circ
f$. The extended tangent space $T{{\mathcal{A}}}_{e}(f)$ to the
${\mathcal{A}}$-orbit ${\mathcal{A}}(f)$ at the point $f$ is the complex
vector space $T{{\mathcal{A}}}_{e}(f)=tf(\theta(n))+wf(\theta{(p)})$. The
extended tangent space $T{\mathcal{L}}_{e}(f)$ to the ${\mathcal{L}}$-orbit of
$f$ is defined as $T{\mathcal{L}}_{e}(f)=wf(\theta{(p)})$.

If ${\mathcal{G}}$ is one of the groups of Mather
${\mathcal{K}},{\mathcal{A}},{\mathcal{L}}$ or ${\mathcal{R}}$, a smooth map-
germ $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ is called
${\mathcal{G}}$-finitely determined (or ${\mathcal{G}}$-finite) if there
exists $k<\infty$ such that for all smooth map-germs
$g:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $j^{k}f(0)=j^{k}g(0)$ we have
that $f$ is ${\mathcal{G}}$-equivalent to $g$, where $j^{k}f(0)$ denotes the
$k$-jet of $f$ at the origin.

Finitely determined map-germs in ${\cal M}_{n}{\cal O}(n,p)$ are particularly
interesting for the class $p\geq 2n$, largely due to the result proved by
Gaffney (see [14, Theorem 2.5]) that if $f$ is not an immersion, then $f$ is
${\mathcal{A}}$-finite if and only if $f$ is ${\cal L}$-finite. In particular,
by Mather-Gaffney geometric criterion of finite determinacy, $f$ is
${\mathcal{A}}$-finite if and only if $0$ is an isolated singularity of $f$.

The above result can be algebraically translated in terms of the ${\cal
G}_{e}$-codimension of $f$, defined as ${\cal
G}_{e}cod(f)=dim_{\mathbb{C}}\frac{\theta(f)}{T{\cal G}_{e}(f)}$, where ${\cal
G}={\mathcal{A}}$ or ${\cal L}$. Furthermore, if $p\geq 2n$ then ${\cal
A}_{e}cod(f)$ is finite if and only if ${\cal L}_{e}cod(f)$ is finite.

In [13] the authors consider the invariant
$\delta_{f}:=dim_{\rm{\mathbb{C}}}\frac{{\cal O}_{n}}{f^{*}({\cal O}_{p})}$
called the delta invariant of $f$. Notice that ${\cal
L}_{e}cod(f)=p\cdot\delta_{f}$. Thus if $p\geq 2n$ the finite determinacy of
$f$ is described in terms of the delta invariant, which is easier to be
calculated than ${\cal A}_{e}cod(f)$.

In what follows we introduce some tools of the theory of curves. For more
details see [6].

Let $\phi_{k}:(\mathbb{C},0)\to(\mathbb{C}^{k},0)$ be a parameterized curve in
$\mathbb{C}^{k}$ in which we assume (up to ${\mathcal{A}}$-equivalence) be
given in the form

$\phi_{k}(y)=(y^{m_{1}},y^{m_{2}}+\sum_{i>m_{2}}a_{i2}y^{i},\ldots,y^{m_{k}}+\sum_{i>m_{k}}a_{ik}y^{i})$
(1)

with $1<m_{1}\leq m_{2}\leq\ldots\leq m_{k}$. We suppose that the map
$\phi_{k}$ is primitive, that is, we cannot reparameterize it by a power of a
new variable and $m_{1}$ is called the multiplicity of the curve.

  * •

Numerical Semigroups and Apéry sequence

A subset $\mathbf{S}\neq\\{0\\}$ is called a semigroup of $\mathbb{N}$ if
$\mathbf{S}$ contains the element $0$ and it is closed under addition. Any
semigroup $\mathbf{S}$ of $\mathbb{N}$ is finitely generated, that is, there
exist $v_{1},\ldots,v_{r}\in\mathbf{S}$ such that $\mathbf{S}:=\langle
v_{1},\ldots,v_{r}\rangle=\\{\lambda_{1}v_{1}+\ldots+\lambda_{r}v_{r};\
\lambda_{i}\in\mathbb{N},i=1,\ldots,r\\}.$ If $gcd(v_{1},\ldots,v_{r})=1$,
then $\mathbf{S}$ has a conductor, which means that there exists an element
$c\in\mathbf{S}$ such that $c-1\notin\mathbf{S}$ and
$c+\mathbb{N}\subset\mathbf{S}$. Consequently $\mathbb{N}\setminus\mathbf{S}$
is finite and its elements are called gaps of $\mathbf{S}$.

In order to describe the gaps of $\mathbf{S}$ we introduce the notion of the
Apéry sequence of $\mathbf{S}$ with respect to an element
$q\in\mathbf{S}\setminus\\{0\\}$, defined inductively by: $a_{0}=0$ and for
$j=1,\ldots,q-1$,

$a_{j}=min\left(\mathbf{S}\setminus\bigcup_{i=0}^{j-1}(a_{i}+q\mathbb{N})\right).$
(2)

  * •

Semigroup and gaps associated to a curve

Let $({\cal C},0)\subset(\mathbb{C}^{k},0)$ be an analytic irreducible curve,
parameterized by $\phi_{k}$ as in (1), with ${\cal
O}:=\mathbb{C}\\{X_{1},\ldots,X_{k}\\}/I$ the local ring of $({\cal C},0)$,
where $I$ is the defining ideal of the curve. The value semigroup $\Gamma$
associated to $\phi_{k}$ (or to the curve) is the set

$\Gamma=\\{ord_{y}(\phi_{k}^{*}(h));\ h\in{\cal O}\setminus\\{0\\}\\}.$

If $\Gamma=\langle v_{1},\ldots,v_{r}\rangle$, since $\phi_{k}$ is a primitive
parameterization, then $gcd(v_{1},\ldots,v_{r})=1$. We consider the Apéry set
$B=\\{a_{0},a_{1},\ldots,a_{m_{1}-1}\\}$ of $\Gamma$ with respect to the
multiplicity $m_{1}$ of $\phi_{k}$. Both the semigroup and the set of gaps $L$
of $\Gamma$ can be determined in terms of the Apéry set $B$. In fact, any
element $s\in\Gamma$ can be written as $s=a+\beta m_{1}$ for some $a\in B$ and
$\beta\in\mathbb{N}\cup\\{0\\}$ and

$L=\left\\{a_{j}-\beta_{j}m_{1};\ j=1,\ldots,m_{1}-1,\
1\leq\beta_{j}\leq\left[\frac{a_{j}}{m_{1}}\right]\right\\},$ (3)

where $[\cdot]$ denotes the integral part. In this case, the delta invariant
$\delta_{\phi_{k}}=dim_{\rm{\mathbb{C}}}\frac{{\cal O}_{1}}{\phi_{k}^{*}({\cal
O}_{k})}$ of $\phi_{k}$ is the number of gaps of $\Gamma$. Thus
$\delta_{\phi_{k}}$ is finite and consequently $\phi_{k}$ is
${\mathcal{A}}$-finitely determined, that is, primitive parameterized curves
are ${\mathcal{A}}$-finite.

##  3 Elementary join maps

In this section we define an “operation” between two map-germs $F$ and $G$,
called the join of $F$ and $G$, in order to produce a new map-germ that
carries, in a certain sense, the $F$ and $G$ structures.

######  Definition 3.1.

Let $F:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p_{1}},0)$ and
$G:(\mathbb{C}^{m},0)\to(\mathbb{C}^{p_{2}},0)$ with $p_{1}\geq n$ and
$p_{2}\geq m$. The join of $F$ and $G$ (with respect to $H$), denoted
$(F*G)_{H}$, is a map from $(\mathbb{C}^{n+m},0)$ to $(\mathbb{C}^{p},0)$ such
that

$(F*G)_{H}(x,y)=(F(x),G(y),H(x,y)),$

for some map-germ $H:(\mathbb{C}^{n+m},0)\to(\mathbb{C}^{p_{3}},0)$ in which
$p=p_{1}+p_{2}+p_{3}$.

Naturally we can extend the above definition for a finite number of maps
$F_{1},\ldots,F_{r}$ with respect to $H$ of the form:
$(F_{1}*\ldots*F_{r})_{H}(x_{1},\ldots,x_{r}):=(F_{1}(x_{1}),\ldots,F_{r}(x_{r}),H(x_{1},\ldots,x_{r}))$.

The ${\mathcal{A}}$-class of $(F*G)_{H}$ is well defined in the sense that if
$F\underset{{\mathcal{A}}}{\sim}\widetilde{F}$ and
$G\underset{{\mathcal{A}}}{\sim}\widetilde{G}$, then
$(F*G)_{H}\underset{{\mathcal{A}}}{\sim}(\widetilde{F}*\widetilde{G})_{\widetilde{H}}$,
for some map-germ $\widetilde{H}$ that is ${\mathcal{A}}$-equivalent to $H$.

Defining some special ${\mathcal{A}}$-finite map-germs which are given as a
join of the identity and a parameterized curve or a join of
${\mathcal{A}}$-finite parameterized curves, we produce new families of
${\mathcal{A}}$-finite map-germs $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$
with $p\geq 2n$.

In the next two results, we provide two families of ${\mathcal{A}}$-finite
maps of corank $1$ and $n$, respectively, that allow us to characterize any
${\mathcal{A}}$-finite monomial map of corank $q$.

######  Proposition 3.2.

Let $\phi_{k}$ be the germ of a parameterized curve in $\mathbb{C}^{k}$ as in
(1) and $I$ the identity in $\mathbb{C}^{n-1}$ with $n,k\geq 2$. Then the join
$(I*\phi_{k})_{H}:(\mathbb{C}^{n},0)\to(\mathbb{C}^{2n+k-2},0)$, with respect
to
$H(\underline{x},y)=(x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y)$,
that is,

$(I*\phi_{k})_{H}(\underline{x},y):=(\underline{x},\ \phi_{k}(y),\
x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y)$

is an ${\mathcal{A}}$-finite map-germ of corank $1$, where
$\underline{x}=(x_{1},\ldots,x_{n-1})$ and $\lambda_{i}\geq 1$ for
$i=1,\ldots,n-1$.

Proof: The proof that $f:=(I*\phi_{k})_{H}$ is ${\mathcal{A}}$-finite follows
a similar idea of [13, Theorem 2.4]. We can verify that $f$ is one-to-one,
since $\phi_{k}$ is a primitive parameterization and in
$\mathbb{C}^{n}\setminus\\{0\\}$ the map $df_{x}$ has maximal rank. Thus $f$
is stable outside the origin, and therefore, ${\mathcal{A}}$-finite by Mather-
Gaffney geometric criterion. However we give an algebraic proof in order to
estimate the delta invariant of $f$.

Let $\Gamma$ be the semigroup of $\phi_{k}$ with conductor $c$ and
$L=\mathbb{N}\setminus\Gamma$ the set of gaps of $\Gamma$. We will prove that
$\delta_{f}$ is finite since the following elements belong to $f^{*}({\cal
O}_{p})$, where $p:=2n+k-2$:

  1. $(a)$

$x_{1}^{r_{1}}\cdots x_{n-1}^{r_{n-1}}y^{s}\in f^{*}({\cal O}_{p})$, for all
$r_{i}\geq 0$ and $s=0$ or $s\geq c\ $;

For each $i\in\\{1,\ldots,n-1\\}$:

  2. $(b)$

$x_{1}^{r_{1}}\cdots x_{i}^{r_{i}}\cdots x_{n-1}^{r_{n-1}}y^{s}\in f^{*}({\cal
O}_{p})$, with $s\in\Gamma$, $s<c\ $, $r_{i}\geq a\lambda_{i}$ for some $a$
depending on $s$ and for all $r_{j}\geq 0$ with $i\neq
j\in\\{1,\ldots,n-1\\}$;

  3. $(c)$

$x_{1}^{r_{1}}\cdots x_{i}^{r_{i}}\cdots x_{n-1}^{r_{n-1}}y^{l}\in f^{*}({\cal
O}_{p})$, with $l\in L$, $r_{i}\geq(m_{1}-1)\lambda_{i}$ and for all
$r_{j}\geq 0$ with $i\neq j\in\\{1,\ldots,n-1\\}$.

In fact, it is clear that $x_{1}^{r_{1}}\cdots x_{n-1}^{r_{n-1}}\in
f^{*}({\cal O}_{p})$ for all $r_{i}\geq 0$. On the other hand, by definition
of the conductor of a semigroup, any integer $s\geq c$ satisfies $s\in\Gamma$.
Thus, there exists $h\in{\cal O}$ such that
$y^{s}=h(\phi_{k})\in\phi_{k}^{*}({\cal O})\subset f^{*}({\cal O}_{p})$, for
all $s\geq c$.

For the item $(b)$, if $s\in\Gamma$ with $s<c$, we only guarantee that there
exists $h\in{\cal O}$ such that $h(\phi_{k})=y^{s}+h.o.t.\in\phi_{k}^{*}({\cal
O})$. But as we remarked before (3), we can write $s=a+\beta m_{1}$, for some
$a$ in the Apéry set $B$ of $\Gamma$ and $\beta\in\mathbb{N}\cup\\{0\\}$. Thus
for each $i=1,\ldots,n-1$ we have

$x_{i}^{a\lambda_{i}}y^{s}=x_{i}^{a\lambda_{i}}y^{a+\beta
m_{1}}=(x_{i}^{\lambda_{i}}y)^{a}(y^{m_{1}})^{\beta}\in f^{*}({\cal O}_{p}).$

In $(c)$, given $l\in L$ it follows by (3) that $l=a-\beta m_{1}$ for some
$a\in B$ and $\beta\in\mathbb{N}$, where $1\leq\beta\leq[\frac{a}{m_{1}}]$.
Since $m_{1}[\frac{a}{m_{1}}]\leq a<m_{1}([\frac{a}{m_{1}}]+1)$ and denoting
$d_{i1}=\lambda_{i}(a-m_{1}[\frac{a}{m_{1}}])$ for each $i=1,\ldots,m_{1}-1$
we have

$x_{i}^{d_{i1}}y^{l}=(x_{i}^{\lambda_{i}}y)^{a-m_{1}[\frac{a}{m_{1}}]}(y^{m_{1}})^{[\frac{a}{m_{1}}]-\beta}\in
f^{*}({\cal O}_{p}).$ (4)

Thus $x_{1}^{r_{1}}\cdots x_{i}^{r_{i}}\cdots x_{n-1}^{r_{n-1}}y^{l}\in
f^{*}({\cal O}_{p})$, for each $l\in L$ and $r_{i}\geq d_{i1}$, where $d_{i1}$
depend on $l\in L$. These computations are sufficient to guarantee that
$\delta_{f}$ is finite. But we will try to obtain a power of $x_{i}$ that does
not depends of the gap. By the inequality $a+1\leq m_{1}([\frac{a}{m_{1}}]+1)$
we have that $d_{i1}\leq(m_{1}-1)\lambda_{i}$ and therefore, by (4), for any
$l\in L$

$x_{i}^{(m_{1}-1)\lambda_{i}}y^{l}=x_{i}^{(m_{1}-1)\lambda_{i}-d_{i1}}\cdot(x_{i}^{d_{i1}}y^{l})\in
f^{*}({\cal O}_{p}).$

In Proposition 3.2, we obtain a family of ${\mathcal{A}}$-finite map-germs
joining the identity and one parameterized curve. A possible question is:
Given two parameterized curves $\phi_{1}$ and $\phi_{2}$, is there a map $H$
such that the map-germ $(\phi_{1}*\phi_{2})_{H}$ is ${\mathcal{A}}$-finite? We
illustrate this problem with an example, which motivates the next result in
which we join a finite number of parameterized curves.

######  Example 3.3.

Let $g(x,y)=(x^{3},x^{4},y^{5},y^{6},x^{2}y,xy^{3})\in{\cal O}(2,6)$ be a
corank $2$ map-germ. The semigroups of the parameterized curves
$(x^{3},x^{4})$ and $(y^{5},y^{6})$ are respectively $\Gamma_{1}=\langle
3,4\rangle$ and $\Gamma_{2}=\langle 5,6\rangle$ with its corresponding set of
gaps $L_{1}=\\{1,2,5\\}$ and $L_{2}=\\{1,\ldots,4,7,8,9,13,14,19\\}$. It is
possible to verify that $\delta_{g}=48$, since a basis to $\frac{{\cal
O}_{2}}{g^{*}({\cal O}_{6})}$ is the set

$\begin{array}[]{l}\gamma=\\{x^{l_{1}},y^{l_{2}},xy,xy^{2},xy^{4},\ldots,xy^{7},xy^{10},xy^{11},xy^{12},xy^{16},xy^{17},xy^{22},x^{2}y^{2},\ldots,x^{2}y^{5},x^{2}y^{8},\vspace{0.1cm}\\\
\hskip
28.45274ptx^{2}y^{9},x^{2}y^{10},x^{2}y^{14},x^{2}y^{15},x^{2}y^{20},x^{3}y,x^{3}y^{2},x^{3}y^{3},x^{3}y^{7},x^{3}y^{8},x^{3}y^{13},x^{4}y,x^{4}y^{4},x^{5}y^{2},\vspace{0.1cm}\\\
\hskip 28.45274ptx^{5}y^{4},x^{6}y^{2},x^{7}y,x^{9}y^{2};\ \mbox{for all}\
l_{i}\in L_{i},\ i=1,2\\}.\end{array}$

With the purpose of constructing an ${\mathcal{A}}$-finite family joining a
finite number of curves, we will introduce some notation: for each
$j=1,\ldots,n$ let $\phi_{k_{j}}:(\mathbb{C},0)\to(\mathbb{C}^{k_{j}},0)$ be a
primitive parameterized curve in $\mathbb{C}^{k_{j}}$, with multiplicity
$m_{j}>1$, given (up to ${\mathcal{A}}$-equivalence) by

$\phi_{k_{j}}(y_{j})=(y_{j}^{m_{j}},y_{j}^{m_{j2}}+\sum_{i>m_{j2}}a_{i2}y_{j}^{i},\ldots,y_{j}^{m_{jk_{j}}}+\sum_{i>m_{jk_{j}}}a_{ik_{j}}y_{j}^{i}),$

with $1<m_{j}:=m_{j1}\leq m_{j2}\leq\ldots\leq m_{jk_{j}}$. If
$\underline{y}:=(y_{1},\ldots,y_{n})$ we denote

$y_{j}^{\mu_{jt}}\underline{\hat{y}}:=(y_{j}^{\mu_{j1}}y_{1},\ldots,y_{j}^{\mu_{j(j-1)}}y_{j-1},y_{j}^{\mu_{j(j+1)}}y_{j+1},\ldots,y_{j}^{\mu_{jn}}y_{n}),$
(5)

where for each $j\in\\{1,\ldots,n\\}$ we have $t\in\\{1,\ldots,n\\}$ with
$t\neq j$ and $\mu_{jt}\geq 1$.

######  Proposition 3.4.

With the previous notations, let
$H:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n(n-1)},0)$ be a map-germ given by
$H(\underline{y})=(y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,\
y_{n}^{\mu_{nt}}\underline{\hat{y}})$ with $n\geq 2$. Then
$(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$,
that is,

$(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}(\underline{y})=(\phi_{k_{1}}(y_{1}),\ldots,\phi_{k_{n}}(y_{n}),y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,\
y_{n}^{\mu_{nt}}\underline{\hat{y}})$

is an ${\mathcal{A}}$-finite map-germ of corank $n$ in which
$p=\sum_{j=1}^{n}k_{j}+n(n-1)$.

Proof: The Mather-Gaffney geometric criterion can be used, as in Proposition
3.2, to prove that $g:=(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}$ is
${\mathcal{A}}$-finite. We also present an algebraic proof in this case. Let
$\Gamma_{j}$ be the semigroup of the parameterized curve $\phi_{k_{j}}$ with
conductor $c_{j}$. We will verify that the following elements are in
$g^{*}({\cal O}_{p})$:

  1. $(a1)$

$y_{1}^{s_{1}}\cdots y_{n}^{s_{n}}\in g^{*}({\cal O}_{p})$ with $s_{j}=0$ or
$s_{j}\geq c_{j}$ for $j\in\\{1,\ldots,n\\}$;

  2. $(b1)$

Given $s_{i}\in\Gamma_{i}$, $s_{i}<c_{i}$, $i\in\\{1,\ldots,n\\}$, then
$y_{k}^{s_{k}}y_{i}^{s_{i}}\in g^{*}({\cal O}_{p})$ for any
$k\in\\{1,\ldots,n\\}$, $k\neq i$, $s_{k}\geq c_{k}+a\mu_{ki}$ with $a$
depending on $s_{i}$;

  3. $(c1)$

Given $l_{i}\in\mathbb{N}\setminus\Gamma_{i}$ with $i\in\\{1,\ldots,n\\}$,
then $y_{k}^{s_{k}}y_{i}^{l_{i}}\in g^{*}({\cal O}_{p})$ for any
$k\in\\{1,\ldots,n\\}$, $k\neq i$, $s_{k}\geq c_{k}+\alpha$ with $\alpha$
depending on $l_{i}$.

The item $(a1)$ follows as in the proof of Proposition 3.2, $(a)$. For $(b1)$
taking $s_{i}=a+\beta m_{i}\in\Gamma_{i}$ with $a$ in the Apéry sequence of
$\Gamma_{i}$ (see the proof of Proposition 3.2, $(b)$) we have
$y_{k}^{c_{k}+a\mu_{ki}}y_{i}^{s_{i}}=y_{k}^{c_{k}}(y_{k}^{\mu_{ki}}y_{i})^{a}(y_{i}^{m_{i}})^{\beta}\in
g^{*}({\cal O}_{p})$.

In $(c1)$, given $l_{i}\in\mathbb{N}\setminus\Gamma_{i}$ we can write
$l_{i}=a-\beta m_{i}$, for some $a$ in the Apéry sequence of $\Gamma_{i}$ and
$1\leq\beta\leq[\frac{a}{m_{i}}]$. Then for any $k\in\\{1,\ldots,n\\}$ with
$k\neq i$ and denoting $d_{ki}=\mu_{ki}(a-m_{i}[\frac{a}{m_{i}}])$ we have (as
in (4)) that

$y_{k}^{c_{k}+d_{ki}}y_{i}^{l_{i}}=y_{k}^{c_{k}}(y_{k}^{\mu_{ki}}y_{i})^{a-m_{i}[\frac{a}{m_{i}}]}(y_{i}^{m_{i}})^{[\frac{a}{m_{i}}]-\beta}\in
g^{*}({\cal O}_{p}).$

In a similar way, if we consider distinct
$i_{1},\ldots,i_{q}\in\\{1,\ldots,n\\}$, $s_{\rho}\in\Gamma_{i_{\rho}}$, with
$s_{\rho}=0$ or $s_{\rho}<c_{i_{\rho}}$, $\rho=1,\ldots,q$, then we can write
$s_{\rho}=a_{\rho}+\beta_{\rho}m_{i_{\rho}}$ with $a_{\rho}$ and
$\beta_{\rho}$ as before. Thus, for any
$k\in\\{1,\ldots,n\\}\setminus\\{i_{1},\ldots,i_{q}\\}$ we have
$y_{k}^{s_{k}}y_{i_{1}}^{s_{1}}\cdots y_{i_{q}}^{s_{q}}\in g^{*}({\cal
O}_{p})$ for all $s_{k}>c_{k}$. More precisely

$y_{k}^{(c_{k}+a_{1}\mu_{ki_{1}}+\ldots+a_{q}\mu_{ki_{q}})}y_{i_{1}}^{s_{1}}\cdots
y_{i_{q}}^{s_{q}}\ =\
y_{k}^{c_{k}}(y_{k}^{\mu_{ki_{1}}}y_{i_{1}})^{a_{1}}\cdots(y_{k}^{\mu_{ki_{q}}}y_{i_{q}})^{a_{q}}(y_{i_{1}}^{m_{i_{1}}})^{\beta_{1}}\cdots(y_{i_{q}}^{m_{i_{q}}})^{\beta_{q}}\in
g^{*}({\cal O}_{p}).$

Finally, given
$l_{\rho}=a_{\rho}-\beta_{\rho}m_{i_{\rho}}\in\mathbb{N}\setminus\Gamma_{i_{\rho}}$,
$\rho=1,\ldots,q$, then for each
$k\in\\{1,\ldots,n\\}\setminus\\{i_{1},\ldots,i_{q}\\}$ and denoting
$d_{ki_{\rho}}=\mu_{ki_{\rho}}(a_{\rho}-m_{i_{\rho}}[\frac{a_{\rho}}{m_{i_{\rho}}}])$,
as in the proof of (c1), $y_{k}^{d_{ki_{\rho}}}y_{i_{\rho}}^{l_{i}}\in
g^{*}({\cal O}_{p})$. Thus

$y_{k}^{(c_{k}+d_{ki_{1}}+\ldots+d_{ki_{q}})}y_{i_{1}}^{l_{1}}\cdots
y_{i_{q}}^{l_{q}}\ =\
y_{k}^{c_{k}}(y_{k}^{d_{ki_{1}}}y_{i_{1}}^{l_{1}})\cdots(y_{k}^{d_{ki_{q}}}y_{i_{q}}^{l_{q}})\in
g^{*}({\cal O}_{p}).$

A general element in $g^{*}({\cal O}_{p})$ can be obtained as a combination of
the previous elements. Thus, $\delta_{g}$ is finite.

######  Definition 3.5.

The maps $(I*\phi_{k})_{H}$ and $(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}$ as in
Proposition 3.2 and Proposition 3.4, respectively, are called elementary
joins.

######  Remark 3.6.

  1. 1.

Notice that by the previous result (module the ${\mathcal{A}}$-action) the
component $H$ of the map-germ $(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}$ may
have some zero components, for example, if some power $\mu_{jt}=1$. Thus, we
can also obtain ${\mathcal{A}}$-finite map-germs in ${\cal O}(n,p)$ for
$p<\sum_{j=1}^{n}k_{j}+n(n-1)$.

  2. 2.

More generally, if $f:=(I*\phi_{k})_{H}$ and
$g:=(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}$ are the elementary join maps, and
$p$ denotes its corresponding target dimension, then we can produce new
${\mathcal{A}}$-finite map-germs, for instance,
$F(\underline{x},y)=(f(\underline{x},y),h(\underline{x},y))$ and
$G(\underline{y})=(g(\underline{y}),k(\underline{y}))$, where $h\in{\cal
O}_{n}/f^{*}({\cal O}_{p})$ and $k\in{\cal O}_{n}/g^{*}({\cal O}_{p})$.

######  Example 3.7.

Let
$f_{a}(x_{1},x_{2},y)=(x_{1},x_{2},y^{4},y^{5},x_{1}y,x_{2}y,ax_{1}x_{2}y^{3})\in{\cal
O}(3,7)$ be a corank $1$ map-germ, where $a=0$ or if $a\neq 0$ then, up to
${\mathcal{A}}$-equivalence, we take $a=1$. In both cases, $f_{a}$ is
${\mathcal{A}}$-finite. The semigroup of the parameterized curve
$(y^{4},y^{5})$ is the set $\Gamma=\langle 4,5\rangle$ and its set of gaps is
$L=\\{1,2,3,6,7,11\\}$. Notice that a basis for $\frac{{\cal
O}_{3}}{f_{a}^{*}({\cal O}_{7})}$ is

$\beta=\\{y^{l},x_{i}y^{2},x_{i}y^{3},x_{i}y^{7},x_{1}x_{2}y^{3},x_{i}^{2}y^{3};\
i=1,2,\forall\ l\in L\\}$

if $a=0$ and $\beta\setminus\\{x_{1}x_{2}y^{3}\\}$ if $a=1$. Thus,
$\delta_{f_{0}}=15$ and $\delta_{f_{1}}=14$.

##  4 General form of ${\mathcal{A}}$-finite monomial maps of corank $q$

Our purpose in this section is to give a general form for any map-germ
$f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p\geq 2n$, satisfying the
properties that $f$ is monomial and ${\mathcal{A}}$-finitely determined.
Moreover we obtain a classification of some classes of parameterized map-
germs.

In the next lemma we point out that the class of parameterized curves appears
naturally in the characterization of ${\mathcal{A}}$-finite monomial maps in
${\cal O}(n,p)$ for $p\geq 2n$.

######  Lemma 4.1.

If $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p\geq 2n$ is a singular
${\mathcal{A}}$-finite monomial map, then there exists at least one
${\mathcal{A}}$-finite parameterized monomial curve
$\phi_{k}:(\mathbb{C},0)\to(\mathbb{C}^{k},0)$ with $k<p$ such that
$f(\underline{x},y)=(f_{1}(\underline{x},y),\ldots,f_{p-k}(\underline{x},y),\phi_{k}(y))$,
where $\underline{x}=(x_{1},\ldots,x_{n-1})$.

Proof: Let $(x_{1},\ldots,x_{n})$ be the variables of an ${\cal L}$-finite
map-germ $f=(f_{1},\ldots,f_{p})$. Thus ${\cal M}_{n}^{\beta}\subset
f^{*}({\cal O}_{p})$ for some $\beta\in\mathbb{N}$, and since $f$ is monomial,
for each $i=1,\ldots,n$, there exists a coordinate of $f$ given by
$x_{i}^{r_{i}}$ for some $r_{i}\geq 1$. Moreover, $f$ singular implies that at
least one variable, namely $x_{n}$, is such that $r_{n}\geq 2$. Let us rename
$x_{n}$ and $r_{n}$ respectively by $y$ and $m_{1}$.

In this way, since $y^{b}\in f^{*}({\cal O}_{p})$ for all $b\geq\beta$ and $f$
is monomial, then there exists $k<p$ such that $f$ admits $k$ components of
the form $f_{j}=y^{m_{j}}$ satisfying $gcd(m_{1},\ldots,m_{k})=1$, or in other
words,
$f(\underline{x},y)=(f_{1}(\underline{x},y),\ldots,f_{p-k}(\underline{x},y),y^{m_{1}},\ldots,y^{m_{k}})$.

In Theorem 4.2 and Corollaries 4.3 and 4.6 we characterize some classes of
${\mathcal{A}}$-finite monomial map-germs in terms of elementary joins.

######  Theorem 4.2.

Let $\phi_{k}(y)=(y^{m_{1}},\ldots,y^{m_{k}})$ be a monomial parameterized
curve in $\mathbb{C}^{k}$, with $1<m_{1}<m_{2}<\ldots<m_{k}$,
$gcd(m_{1},\ldots,m_{k})=1$ and $n,k\geq 2$. With the notations of Proposition
3.2, a singular monomial map-germ
$f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{2n+k-2},0)$ satisfying
$f(\underline{0},y)=(0,\ldots,0,\phi_{k}(y))$ is ${\mathcal{A}}$-finite if and
only if $f$ is ${\mathcal{A}}$-equivalent to

$(I*\phi_{k})_{H}(\underline{x},y)=(\underline{x},y^{m_{1}},\ldots,y^{m_{k}},x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y).$

Proof: Notice that the sufficient condition follows from Proposition 3.2.

By hypothesis
$f(\underline{x},y)=(f_{1}(\underline{x},y),\ldots,f_{2n-2}(\underline{x},y),\phi_{k}(y))$.
From the proof of Lemma 4.1, there exist $n-1$ components of $f$ of the form
$x_{i}^{r_{i}}$ with $r_{i}\geq 1$. If some $r_{i}>1$, namely $r_{1}$, the
above lemma guarantees that $f$ has as coordinates a parameterized curve in
the variable $x_{1}$. Let us suppose, without loss of generality, that $f$
admits components: $x_{1}^{n_{1}},x_{1}^{n_{2}},y^{m_{1}},\ldots,y^{m_{k}}$,
where $1<n_{1}<n_{2}$ and $gcd(n_{1},n_{2})=1$. Since $f$ is monomial and
$\delta_{f}$ is finite, $f^{*}({\cal O}_{2n+k-2})$ must contain the elements
$x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y,x_{2}^{s_{2}}x_{1},\ldots,x_{n-1}^{s_{n-1}}x_{1},y^{s_{1}}x_{1}$,
for some $\lambda_{i},s_{i}\in\mathbb{N}$, $i=1,\ldots,n-1$. But this would
imply that $p>2n+k-2$.

If we suppose, for instance, that $f$ has components:
$x_{1}x_{2},x_{2},\ldots,x_{n-1},y^{m_{1}},\ldots,y^{m_{k}}$, then it follows
that $f^{*}({\cal O}_{2n+k-2})$ must contain the elements:
$x_{1}^{s_{1}},x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y$ for some
$s_{1},\lambda_{i}\in\mathbb{N}$, $i=1,\ldots,n-1$. If $s_{1}>1$, then $f$
admits a parameterized curve in the variable $x_{1}$. Thus $f$ also need to
have as components (at least):
$x_{1}^{s_{2}},x_{3}^{s_{3}}x_{1},\ldots,x_{n-1}^{s_{n-1}}x_{1}$ for some
$s_{j}\in\mathbb{N}$ with $j=2,\ldots,n-1$ and $gcd(s_{1},s_{2})=1$. By the
same argument as above, we get a contradiction. Thus $s_{1}=1$. With a similar
analysis for the other cases, we conclude that $f$ is a corank one map. In
fact, it follows that $f$ is ${\mathcal{A}}$-equivalent to a join of the
identity $I$ in $\mathbb{C}^{n-1}$ and $\phi_{k}$, with respect to the map
$H:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n-1},0)$,
$H(\underline{x},y)=(x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y)$.

It is interesting to observe that if $f\in{\cal M}_{n}{\cal O}(n,2n+k-2)$ is a
singular monomial and ${\mathcal{A}}$-finite map-germ, then, by a dimension
argument, $f$ cannot have as component a parameterized curve
$\phi_{j}:(\mathbb{C},0)\to(\mathbb{C}^{j},0)$ for $j>k$.

We can ask whether a result, as we proved in Theorem 4.2, can be obtained for
a map-germ $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+k},0)$ with
$k=1,\ldots,n-1$. We believe that the answer is no. The only class we got in
${\cal O}(n,n+k)$ satisfying that the germ is ${\mathcal{A}}$-finite and
monomial was the immersion and

$F(\underline{x},y)=(\underline{x},x_{i_{1}}y,x_{i_{2}}y,\ldots,x_{i_{k}}y,y^{2}),\
$

for each $k=1,\ldots,n-1$ and distinct $i_{1},\ldots
i_{k}\in\\{1,\ldots,n-1\\}$.

As a consequence of Lemma 4.1 and Theorem 4.2, we extend for maps in ${\cal
M}_{n}{\cal O}(n,2n)$ the classification of parameterized monomial surfaces in
$\mathbb{C}^{4}$ obtained in [13, Theorem 2.4].

######  Corollary 4.3.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{2n},0)$ be a monomial map-germ with
$n\geq 2$. Then, $f$ is ${\mathcal{A}}$-finite if and only if $f$ is
${\mathcal{A}}$-equivalent to

  1. (i)

$(\underline{x},y,0,\ldots,0)$ (immersion);

  2. (ii)

$(\underline{x},y^{m_{1}},y^{m_{2}},x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y),$
where $\lambda_{i}\geq 1$ for $i=1,\ldots,n-1$ and $gcd(m_{1},m_{2})=1$.

In what follows, we give a general form of an ${\mathcal{A}}$-finite monomial
map in ${\cal O}(n,p)$ of corank $q$, for $p\geq 2n$. For each $j=1,\ldots,q$
the map
$\phi_{k_{j}}(y_{j})=(y_{j}^{m_{j}},y_{j}^{m_{j2}},\ldots,y_{j}^{m_{jk_{j}}})$
denotes a primitive parameterized curve in $\mathbb{C}^{k_{j}}$ with
$1<m_{j}:=m_{j1}<m_{j2}<\ldots<m_{jk_{j}}$ and if
$\underline{x}:=(x_{1},\ldots,x_{n-q})$ and
$\underline{y}:=(y_{1},\ldots,y_{q})$, then
$\underline{x}^{\lambda_{ij}}y_{j}:=(x_{1}^{\lambda_{1j}}y_{j},\ldots,x_{n-q}^{\lambda_{(n-q)j}}y_{j})$
and $y_{j}^{\mu_{jt}}\hat{\underline{y}}$ is given as in (5), where
$\lambda_{ij}\geq 1$, $\mu_{jt}\geq 1$ for $i\in\\{1,\ldots,n-q\\}$,
$j,t\in\\{1,\ldots,q\\}$ with $t\neq j$ and $I$ denotes the identity map in
$\mathbb{C}^{n-q}$.

######  Theorem 4.4.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ be a singular corank $q$
monomial map-germ with $p\geq 2n$. Then $f$ is ${\mathcal{A}}$-finite if and
only if $f$ is ${\mathcal{A}}$-equivalent to

$(I*\phi_{k_{1}}*\ldots*\phi_{k_{q}})_{H}(\underline{x},\underline{y})=(I(\underline{x}),\phi_{k_{1}}(y_{1}),\ldots,\phi_{k_{q}}(y_{q}),H(\underline{x},\underline{y})),$
(6)

where
$H(\underline{x},\underline{y})=(\underline{x}^{\lambda_{i1}}y_{1},\ldots,\underline{x}^{\lambda_{iq}}y_{q},y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,y_{q}^{\mu_{qt}}\underline{\hat{y}},h(\underline{x},\underline{y}))$
for some monomial map $h(\underline{x},\underline{y})$ such that $h$ is zero
on each coordinate axis. In particular, if all $\mu_{jt}=1$ the map $H$ can
given by

$H(\underline{x},\underline{y})=(\underline{x}^{\lambda_{i1}}y_{1},\ldots,\underline{x}^{\lambda_{iq}}y_{q},y_{1}y_{2},\ldots,y_{1}y_{q},y_{2}y_{3},\ldots,y_{2}y_{q},\ldots,y_{q-1}y_{q},h(\underline{x},\underline{y})).$

Proof: It follows from Propositions 3.2 and 3.4 that the map (6) is
${\mathcal{A}}$-finite.

Conversely, if $f$ is a corank $q$ map-germ, up to change of coordinates, we
can assume that
$f(\underline{x},\underline{y})=(\underline{x},f_{n-q+1}(\underline{x},\underline{y}),\ldots,f_{p}(\underline{x},\underline{y}))$.
Since $f$ is ${\mathcal{A}}$-finite and monomial, it follows from Lemma 4.1,
that for each $j=1,\ldots,q$ there exists $k_{j}$ with $1<k_{j}\leq p-(n-q)$
such that $f$ has components $\phi_{k_{j}}(y_{j})$ as previously defined. If
$m_{j}=1$ for some $j=1,\ldots,q$, then the corank of $f$ would be less than
$q$. Thus, $f$ is ${\mathcal{A}}$-equivalent to
$(I*\phi_{k_{1}}*\ldots*\phi_{k_{q}})_{H}$ for some map $H$. Similarly as in
the proof of Theorem 4.2, if $f$ is monomial and $\delta_{f}$ is finite, the
map $H$ has to be given by
$H(\underline{x},\underline{y})=(\underline{x}^{\lambda_{i1}}y_{1},\ldots,\underline{x}^{\lambda_{iq}}y_{q},y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,y_{q}^{\mu_{qt}}\underline{\hat{y}},h(\underline{x},\underline{y}))$,
for some monomial map $h$ that, up to ${\mathcal{A}}$-equivalence, is zero on
each coordinate axis.

######  Remark 4.5.

Notice that in Theorem 4.4, the components of $h$ can be zero. Furthermore, if
$\phi_{k_{j}}$ or $h$ are not monomials, we still have that the map (6) is
${\mathcal{A}}$-finite, that is, combining elementary joins we produce a lot
of ${\mathcal{A}}$-finite maps.

Particular interesting cases in Theorem 4.4 are when $f$ is one of the
elementary join maps:

$\begin{array}[]{l}\bullet\ corank\ 1:\
(\underline{x},y^{m_{1}},\ldots,y^{m_{r}},x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y);\vspace{0.2cm}\\\
\bullet\ corank\ n:\ (\phi_{k_{1}}(y_{1}),\ldots,\phi_{k_{n}}(y_{n}),\
y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,y_{n}^{\mu_{nt}}\underline{\hat{y}}).\end{array}$

######  Corollary 4.6.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ be a singular corank $q$
monomial map-germ, with $p\geq 2n.$ If $f$ is ${\mathcal{A}}$-finite, then
$p\geq n(q+1)-q(q-1)/2$. In particular, if $p=n(q+1)-q(q-1)/2$, then $f$ is
${\mathcal{A}}$-finite if and only if

$f\ \underset{{\mathcal{A}}}{\sim}\ (I*\phi_{k_{1}}*\ldots*\phi_{k_{q}})_{H},$

where for each $j=1,\ldots,q$ the map
$\phi_{k_{j}}(y_{j})=(y_{j}^{m_{j}},y_{j}^{m_{j1}})$ is a primitive
parameterized plane curve and
$H(\underline{x},\underline{y})=(\underline{x}^{\lambda_{i1}}y_{1},\ldots,\underline{x}^{\lambda_{iq}}y_{q},y_{1}y_{2},\ldots,y_{1}y_{q},y_{2}y_{3},\ldots,y_{2}y_{q},\ldots,y_{q-1}y_{q})$.

By the previous result, in order to have an ${\mathcal{A}}$-finite monomial
map-germ in ${\cal M}_{n}{\cal O}(n,p)$ of corank $n$, then
$p\geq\frac{n(n+3)}{2}$. But for corank $1$ maps, we need $p\geq 2n$.

######  Example 4.7.

Let $h(x,y,z)=(x,y^{2},y^{3},z^{2},z^{3},xy,xz,yz)\in{\cal O}(3,8)$ be a
corank $2$ map-germ. In this case, a basis for $\frac{{\cal
O}_{3}}{h^{*}({\cal O}_{8})}$ is $\\{y,z,y^{2}z,yz^{2}\\}$ and thus
$\delta_{h}=4$.

##  5 Estimating some analytical invariants

In this section, given $f$ in ${\cal O}(n,p)$ with $p\geq 2n$ we estimate the
${\mathcal{A}}_{e}$-codimension and the delta invariant of $f$, for the
classes of map-germs of the previous sections. When $p=2n$, we obtain some
formulas for the number of double points in a stable perturbation of $f$. In
the last subsection, we study these invariants under some projection of $f$.
This approach can be useful when the projection has good properties, such as
being stable.

If $f\in{\cal M}_{n}{\cal O}(n,p)$ with $p\geq 2n$ satisfies
$\delta_{f}<\infty$, then $f$ is ${\mathcal{A}}$-finite. Moreover,

${\cal A}_{e}cod(f)\leq{\mathcal{L}}_{e}cod(f)=p\cdot\delta_{f}.$

In this way, it is interesting to obtain some estimates for $\delta_{f}$.

Let $F:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p_{1}},0)$,
$G:(\mathbb{C}^{m},0)\to(\mathbb{C}^{p_{2}},0)$ and
$H:(\mathbb{C}^{n+m},0)\to(\mathbb{C}^{p_{3}},0)$ with $p_{1}\geq n$,
$p_{2}\geq m$ such that
$K=(F*G)_{H}:(\mathbb{C}^{n+m},0)\to(\mathbb{C}^{p},0)$ is a join of $F$ and
$G$ with respect to $H$, that is, $K(x,y)=(F(x),G(y),H(x,y))$ where
$p=p_{1}+p_{2}+p_{3}$.

Supposing $p_{1}\geq 2n$, $p_{2}\geq 2m$, $\delta_{F}$ and $\delta_{G}$
finite, if $H$ is zero on each coordinate axis, then it is easy to see that

$\delta_{K}\geq\delta_{F}+\delta_{G}.$ (7)

In order to estimate some invariants of $(I*\phi_{k})_{H}$ in Proposition 3.2,
we consider the following notations: the integers
$\lambda_{\min}:=\underset{1\leq i\leq n-1}{\min}\\{\lambda_{i}\\}$, $\
\lambda:=(m_{1}-1)\sum_{i=1}^{n-1}\lambda_{i}-n+1$,

$\begin{array}[]{l}\kappa_{1}:=\displaystyle\sum_{i=1}^{n-1}\lambda_{i}+\scriptsize{\sum_{j=1}^{\lambda_{\min}}\left(\begin{array}[]{c}n+j-2\\\
j\\\ \end{array}\right)}-(n-1)\lambda_{\min}-n+2,\\\
\kappa_{2}:=\displaystyle\sum_{j=1}^{\lambda}\scriptsize{\left(\begin{array}[]{c}n+j-2\\\
j\\\ \end{array}\right)}+1\ \ \ and\ \ \
\kappa_{3}:=\displaystyle\sum_{i=1}^{n-1}\lambda_{i}-n.\end{array}$

######  Proposition 5.1.

Let $f(\underline{x},y)=(\underline{x},\ \phi_{k}(y),\
x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y)$ as in Proposition 3.2.
Then

  1. (i)

$\kappa_{1}\delta_{\phi_{k}}\leq\delta_{f}\leq\kappa_{2}\delta_{\phi_{k}},$

  2. (ii)

$(n-1)\kappa_{3}\delta_{\phi_{k}}+k(m_{1}-2)(\kappa_{3}+2)\leq{\mathcal{A}}_{e}cod(f)\leq(2n+k-2)\kappa_{2}\delta_{\phi_{k}}$.

Proof: Denote $p=2n+k-2$ and $c$ the conductor of the semigroup $\Gamma$ of
$\phi_{k}$. It follows from the proof of Proposition 3.2 that
$x_{i}^{r_{i}},y^{b},x_{i}^{p_{i}}y^{s},x_{i}^{q_{i}}y^{l}\in f^{*}({\cal
O}_{p})$, for all $r_{i}\in\mathbb{N}$, $b\geq c$, $s\in\Gamma$ for some
$p_{i}$ and $l\in L=\mathbb{N}\setminus\Gamma$ and
$q_{i}\geq(m_{1}-1)\lambda_{i}$, with $i=1,\ldots,n-1$. Now we will analyze
the element $y^{s}$ with $s\in\Gamma$ and $s$ less than the conductor $c$. If
$s\in\Gamma$, then there exists $h\in{\cal O}$ such that
$h\circ\phi_{k}(y)=y^{s}+a_{s+1}y^{s+1}+\ldots\in\phi_{k}^{*}({\cal
O}_{k})\subset f^{*}({\cal O}_{p})$ with $a_{j}\in\mathbb{C}$ for
$j=s+1,\ldots$. For each $j\in\Gamma\setminus\\{0\\}$ with $j>s$ in the
expansion $h\circ\phi_{k}(y)$, there exists $h_{j}\in{\cal O}$ such that
$h_{j}\circ\phi_{k}(y)=b_{j}y^{j}+\ldots$, with $b_{j}\neq 0$. Thus

$\overline{h}=h-\frac{a_{j}}{b_{j}}\sum_{j\in\Gamma}h_{j}\in{\cal O}$

satisfy
$\overline{h}\circ\phi_{k}(y)=y^{s}+a_{l_{1}}y^{l_{1}}+\ldots+a_{l_{t}}y^{l_{t}}$
for some finite $l_{1},\ldots,l_{t}\in L$ and $a_{l_{i}}\in\mathbb{C}$, since
$y^{b}\in f^{*}({\cal O}_{p})$ for $b\geq c$. Denoting $\beta$ a basis of
${\cal O}_{n}/f^{*}({\cal O}_{p})$, we have that $y^{l}\in\beta$ for all $l\in
L$. In this way, by the expression of $\overline{h}\circ\phi_{k}$, we do not
consider in $\beta$ the element $y^{s}$, with $s\in\Gamma$. With a similar
analysis $x_{1}^{r_{1}}\ldots x_{n-1}^{r_{n-1}}y^{s}\notin\beta$ for
$r_{i}\geq 0$ and $s\in\Gamma$.

Thus $\beta$ is contained in $\\{y^{l},x_{1}^{r_{1}}\ldots
x_{n-1}^{r_{n-1}}y^{l};\ \forall\ l\in L,1\leq d\leq\lambda\\}$ with
$d=\sum_{i=1}^{n-1}r_{i}$ and
$\lambda:=(m_{1}-1)\sum_{i=1}^{n-1}\lambda_{i}-n+1.$ Therefore

$\delta_{f}\leq\delta_{\phi_{k}}+\sharp\\{x_{1}^{r_{1}}\ldots
x_{n-1}^{r_{n-1}}y^{l};\ \forall\ l\in L,1\leq
d\leq\lambda\\}=\kappa_{2}\delta_{\phi_{k}}.$

On the other hand, the elements $y^{l},x_{i}^{r_{i}}y^{l}\in\beta$ for all
$1\leq r_{i}\leq\lambda_{i}-1$, $l\in L$ and $i=1,\ldots,n-1$. Moreover
$x_{1}^{r_{1}}\ldots x_{n-1}^{r_{n-1}}y^{l}\in\beta$ for all $l\in L$ such
that $1\leq d\leq\lambda_{\min}$, eliminating the monomials
$x_{1}^{d}y^{l},\ldots,x_{n-1}^{d}y^{l}$ that have been considered in $\beta$.
Thus,

$\delta_{f}\geq\sharp\\{y^{l},x_{i}^{r_{i}}y^{l};\ \forall\ l\in L,1\leq
r_{i}\leq\lambda_{i}-1\\}+\sharp\\{x_{1}^{r_{1}}\ldots
x_{n-1}^{r_{n-1}}y^{l};\forall\ l\in L,1\leq
d\leq\lambda_{\min}\\}-(n-1)\lambda_{\min}\delta_{\phi_{k}}$

and we obtain $\delta_{f}\geq\kappa_{1}\delta_{\phi_{k}}$.

The upper bound of ${\mathcal{A}}_{e}cod(f)$ is obtained from inequality
${\cal A}_{e}cod(f)\leq p\delta_{f}$ and item $(i)$. Now, we denote by $e_{i}$
the $i$-th element of the canonical basis of $\theta(p)$. For the lower bound
of the ${\mathcal{A}}_{e}$-codimension of $f$, note that the following sets
are not contained in $T{\mathcal{A}}_{e}(f)$:

$\\{\\{x_{i}^{r_{i}}y^{l}\\}e_{n+j-1};\ i,j\in\\{1,\ldots,n-1\\},\forall\ l\in
L,1\leq r_{i}\leq\lambda_{i}-2\ \mbox{if}\ i=j\ \mbox{and}\ 1\leq
r_{i}\leq\lambda_{i}-1,\ \mbox{if}\ i\neq j\\}$

and

$\\{\\{y^{l},x_{i}^{r_{i}}y^{l}\\}e_{j};\
i\in\\{1,\ldots,n-1\\},j\in\\{2n-1,\ldots,2n+k-2\\},1\leq
r_{i}\leq\lambda_{i}-1\ \mbox{and}\ 1\leq l\leq m_{1}-2\\}.$

For the map $(\phi_{k_{1}}*\ldots*\phi_{k_{n}})_{H}$ in Proposition 3.4 the
estimate for the delta invariant is more general than the previous result. If
$\Gamma_{j}$ is the semigroup of $\phi_{k_{j}}$ with conductor $c_{j}$, we
denote $c=\sum_{j=1}^{n}c_{j}$.

######  Proposition 5.2.

Let $g(\underline{x},y)=(\phi_{k_{1}}(y_{1}),\ldots,\phi_{k_{n}}(y_{n}),\
y_{1}^{\mu_{1t}}\underline{\hat{y}},\ldots,\
y_{n}^{\mu_{nt}}\underline{\hat{y}})$ as in Proposition 3.4. Then

$\sum_{j=1}^{n}\delta_{\phi_{k_{j}}}\ \leq\ \delta_{g}\ \leq\
\sum_{j=1}^{c}\left(\begin{array}[]{c}n+j-1\\\ j\\\ \end{array}\right).$

Proof: It follows from (7) the lower estimate for $\delta_{g}$. On the other
hand, if $\gamma$ denotes a basis for ${\cal O}_{n}/g^{*}({\cal O}_{p})$, by
the proof of Proposition 3.4, $\gamma\subset\\{y_{1}^{s_{1}}\cdots
y_{n}^{s_{n}};\ 1\leq d\leq c,d=\sum_{j=1}^{n}s_{j}\\}$. Thus, we obtain the
result.

A map-germ $f$ is called ${\mathcal{A}}$-stable if $\theta(f)=T{\cal
A}_{e}(f)$. Similarly as in [13, Proposition 2.1], it is possible to prove for
$p\geq 2n$ that $f$ is ${\mathcal{A}}$-stable if and only if $\delta_{f}=0$.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{2n},0)$ be an ${\mathcal{A}}$-finite
map. We recall that a stable perturbation $f_{t}:U\to(\mathbb{C}^{2n},0)$ of
$f$, where $U$ is a small neighborhood of the origin, has a finite number
$d(f)$ of transverse double points (see [5]).

If $f$ is an ${\mathcal{A}}$-finite corank one map-germ, it is well known that
up to change of coordinates $f$ is given by
$f(\underline{x},y)=(\underline{x},f_{n}(\underline{x},y),\ldots,f_{2n}(\underline{x},y))$,
where $\underline{x}:=(x_{1},\ldots,x_{n-1})$. In this case, the number of
double points of $f$ is

$d(f):=\displaystyle\frac{1}{2}dim_{\mathbb{C}}\ \frac{{\cal
O}_{n+1}}{\varphi^{*}({\cal M}_{n+1}){\cal O}_{n+1}},$ (8)

where $\varphi:(\mathbb{C}^{n+1},0)\to(\mathbb{C}^{n+1},0)$,
$(\underline{x},y,z)\mapsto(\varphi_{n},\ldots,\varphi_{2n})$ with
$\varphi_{i}(\underline{x},y,z):=\frac{f_{i}(\underline{x},z)-f_{i}(\underline{x},y)}{z-y}$,
for each $i=n\ldots,2n$. Moreover, if $f$ is quasihomogeneous of type
$(\omega_{1},\ldots,\omega_{n},d_{n},\ldots,d_{2n})$, where
$\omega_{1},\ldots,\omega_{n-1},\omega_{n}$ are the weights of the variables
$x_{1},\ldots,x_{n-1},y$ and $d_{n},\ldots,d_{2n}$ are the weighted degrees of
the components functions $f_{n},\ldots,f_{2n}$ of $f$, respectively, then

$d(f)=\frac{\prod_{i=n}^{2n}(d_{i}-\omega_{n})}{2\omega_{1}\ldots\omega_{n-1}\omega_{n}^{2}}.$
(9)

######  Proposition 5.3.

Let $\phi_{2}(y)=(y^{m_{1}},y^{m_{2}})$ with $1<m_{1}<m_{2}$,
$gcd(m_{1},m_{2})=1$ and for each $i=1,2$ the map
$h_{i}(\underline{x},y)=y^{m_{i}}+\sum_{j>m_{i}}a_{ij}(\underline{x})y^{j}$
satisfies $a_{ij}(\underline{x})\in{\cal M}_{n-1}$.

  1. 1.

If
$f(\underline{x},y)=(\underline{x},\underline{x}y,h_{1}(\underline{x},y),h_{2}(\underline{x},y))$,
then $d(f)=\frac{(m_{1}-1)(m_{2}-1)}{2}=\delta_{\phi_{2}}$.

  2. 2.

If
$f(\underline{x},y)=(\underline{x},y^{m_{1}},y^{m_{2}},x_{1}^{\lambda_{1}}y,\ldots,x_{n-1}^{\lambda_{n-1}}y)$,
then $d(f)=\left(\prod_{i=1}^{n-1}\lambda_{i}\right)\delta_{\phi_{2}},$ for
$\lambda_{i}\geq 1$.

Proof: We can see $f$ as a deformation of the map-germ
$f^{\prime}(\underline{x},y)=(\underline{x},\underline{x}y,y^{m_{1}},y^{m_{2}})$
which is ${\mathcal{A}}$-finite by Proposition 3.2. Since the
${\mathcal{A}}_{e}$-codimension is an upper-semicontinuous invariant, that is,
for all sufficiently small values of deformation parameters, we have
${\mathcal{A}}_{e}cod(f)\leq{\mathcal{A}}_{e}cod(f^{\prime})<\infty$, we
conclude that $f$ is an ${\mathcal{A}}$-finite germ.

By definition of $d(f)$ in (8), $\varphi_{i}(\underline{x},y,z)=x_{i-n+1}$ for
$i=n,\ldots,2n-2$, and denoting $P_{k}(z,y)=\frac{z^{k+1}-y^{k+1}}{z-y}$ we
have

$\varphi_{2n+i-2}(\underline{x},y,z)=P_{m_{i}-1}(z,y)+\sum_{j>m_{i}}a_{ij}(\underline{x})P_{j-1}(z,y)$

for $i=1,2$.

We will prove that if $r,s\in\mathbb{N}$ with $gcd(r,s)=1$ then the ideal
$\langle P_{r-1}(z,y),P_{s-1}(z,y)\rangle$ in ${\cal O}_{z,y}$ has finite
codimension. In fact
$(z-y)P_{r-1}(z,y)=z^{r}-y^{r}=\prod_{i=0}^{r-1}(z-\omega^{i}y)$ and
$(z-y)P_{s-1}(z,y)=z^{s}-y^{s}=\prod_{i=0}^{s-1}(z-\upsilon^{i}y),$ with
$\omega$ (resp. $\upsilon$) $r$-th (resp. $s$-th) primitive root of unity.
Notice that $\eta$ is a $d$-th root of unity, with $d=gcd(r,s)$ if and only if
$\eta$ is a $r$-th and $s$-th root of unity, that is, the polynomials
$z^{r}-y^{r}$ and $z^{s}-y^{s}$ have $d$ common roots. In this way by
hypothesis $(z-y)P_{r-1}(z,y)=(z-y)f_{1}$ and $(z-y)P_{s-1}(z,y)=(z-y)f_{2},$
where $f_{1}$ and $f_{2}$ don’t have common roots. Therefore $P_{r-1}$ and
$P_{s-1}$ have no common components which implies that the codimension of the
ideal generated by them is finite.

Since

$d(f)=\frac{1}{2}dim_{\mathbb{C}}\frac{{\cal
O}_{\underline{x},y,z}}{\langle\underline{x},\varphi_{2n-1},\varphi_{2n}\rangle}=\frac{1}{2}dim_{\mathbb{C}}\frac{{\cal
O}_{y,z}}{\langle P_{m_{1}-1}(z,y),P_{m_{2}-1}(z,y)\rangle}$

in which $gcd(m_{1},m_{2})=1$, we have that the polynomials
$P_{m_{i}-1},i=1,2$ have no common factor, then $d(f)$ is finite and by
Bézout’s Theorem we have the required formula, remembering that the delta
invariant of $\phi_{2}$ is $\delta_{\phi_{2}}=\frac{(m_{1}-1)(m_{2}-1)}{2}$.

For the item $2$, the formula for $d(f)$ can be easily obtained by (9).

###  5.1 ${\mathcal{A}}$-invariants under projections

For the study of invariants of a map $f$ under some projection of it, we only
assume $f\in{\cal M}_{n}{\cal O}(n,p)$ in which $p\geq n$ and $0$ is an
isolated singularity of $f$.

Fixed $k\in\\{1,\dots,p-1\\}$ let $\pi_{k}:\mathbb{C}^{p}\to\mathbb{C}^{k}$ be
a canonical projection, and $g_{k}=\pi_{k}\circ f$ such that
$f\underset{{\mathcal{A}}}{\sim}(g_{k},h),$ where
$h:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p-k},0)$. Notice that
$\theta(f)\simeq\theta(g_{k})\oplus\theta(h)$. It is well known that
$\omega\in T{\cal A}_{e}(f)$ if and only if there exist $\epsilon\in\theta(n)$
and $\eta\in\theta(p)$ such that $\omega=df(\epsilon)+\eta(f)$.

We consider the subspaces

$T_{k}=\left\\{\omega_{k}\in\theta(g_{k});\
\omega_{k}=dg_{k}(\epsilon)+(\pi_{k}\circ\eta)(f),\
\epsilon\in\theta(n),\eta\in\theta(p)\right\\}\ \mbox{and}$

$T=\\{\omega_{h}\in\theta(h);\ (0,\omega_{h})\in T{\cal A}_{e}(f)\\}$

and their codimensions are respectively defined as $cod\
T_{k}=\dim_{\rm{\mathbb{C}}}\displaystyle\frac{\theta(g_{k})}{T_{k}}$ and
$cod\ T=\dim_{\rm{\mathbb{C}}}\displaystyle\frac{\theta(h)}{T}.$

######  Proposition 5.4.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p\geq n$, such that
$f\underset{{\mathcal{A}}}{\sim}(g_{k},h)$ as above, for some $k=1,\dots,p-1$.
If $cod\ T_{k}$ and $cod\ T$ are finite, then $f$ is ${\mathcal{A}}$-finitely
determined and

${\cal A}_{e}cod(f)=cod\ T_{k}+cod\ T.$

Proof: The proof is similar to the Proposition 2.1 in [9] for divergents
diagrams. If $\omega\in\theta(f)$, we denote by $\overline{\omega}$ the class
of this element in the quotient $\theta(f)/T{\cal A}_{e}(f)$, similarly (with
an abuse of notation) for $\theta(g_{k})$ and $\theta(h)$. The result follows
considering the following exact sequence:

$0\ \longrightarrow\ \frac{\theta(h)}{T}\ \stackrel{{\scriptstyle
i^{*}}}{{\longrightarrow}}\ \frac{\theta(f)}{T{\cal A}_{e}(f)}\
\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}\
\frac{\theta(g_{k})}{T_{k}}\ \longrightarrow\ 0,$

where if $\overline{\omega_{h}}\in\theta(h)/T$ then
$i^{*}(\overline{\omega_{h}})=\overline{(0,\omega_{h})}$ and given
$\omega=(\omega_{1},\ldots,\omega_{p})\in\theta(f)$ the projection
$\pi^{*}(\overline{\omega})=\overline{(\omega_{1},\dots,\omega_{k})}$.

In a natural way, we can extend the above result for multigerms.

For the germ $h:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p-k},0)$ we define the map
$\overline{w}h:\theta(p)\to\theta(h)$ by $\overline{w}h(\eta)=(\eta_{k+1}\circ
f,\dots,\eta_{p}\circ f)$, where
$\eta:=(\eta_{1},\ldots,\eta_{p})\in\theta(p)$. Given $\omega_{h}\in T$ we
have $(0,\omega_{h})\in T{\cal A}_{e}(f)$, which implies that there exist
$\epsilon\in\theta(n)$ and $\eta\in\theta(p)$ such that
$dg_{k}(\epsilon)+(\pi_{k}\circ\eta)(f)=0$ and
$dh(\epsilon)+\overline{w}h(\eta)=\omega_{h}$. Let

$\rho:\frac{\theta(h)}{\overline{\omega}h(\theta(p))}\longrightarrow\frac{\theta(h)}{T}$
(10)

such that $\rho$ of the class of $\omega_{h}\in\theta(h)$ in
$\theta(h)/\overline{\omega}h(\theta(p))$ is the class of this element in
$\theta(h)/T$.

In the next result we present a relation among ${\cal A}_{e}cod(f)$ and
invariants of some projection $g_{k}$ of $f$, that can be interesting if you
have some information about $g_{k}$.

######  Theorem 5.5.

Consider $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ with $p\geq 2n$ and
$f\underset{{\mathcal{A}}}{\sim}(g_{k},h)$. If $g_{k}$ is
${\mathcal{A}}$-finitely determined and $\delta_{f}$ is finite, then

  1. (i)

${\cal A}_{e}cod(f)\leq{\mathcal{A}}_{e}cod(g_{k})+(p-k)\delta_{f}.$

  2. (ii)

If $g_{k}$ is stable, then ${\cal A}_{e}cod(f)=cod\
T=(p-k)\delta_{f}-dim_{\mathbb{C}}Ker(\rho)$.

Proof: $(i)$ Notice that $\theta(h)/T$ is isomorphic to a submodule of
$\bigoplus_{j=1}^{p-k}{\cal O}_{n}/f^{*}({\cal O}_{p})$ and therefore $cod\
T\leq(p-k)\delta_{f}<\infty$. On the other hand, the extended tangent space to
$g_{k}$ is

$T{\cal
A}_{e}(g_{k})=\\{\omega_{k}\in\theta(g_{k});\omega_{k}=dg_{k}(\epsilon)+\eta(g_{k}),\epsilon\in\theta(n),\eta\in\theta{(k)}\\},$

which implies $cod\ T_{k}\leq{\mathcal{A}}_{e}cod(g_{k})$. By Proposition 5.4,
${\cal A}_{e}cod(f)\leq{\mathcal{A}}_{e}cod(g_{k})+(p-k)\delta_{f}.$

$(ii)$ If $g_{k}$ is stable, then ${\mathcal{A}}_{e}cod(g_{k})=0$ and
consequently $cod\ T_{k}=0$. Thus ${\cal A}_{e}cod(f)=cod\ T$. Considering the
map $\rho$ defined in (10) it follows by exact sequence

$0\ \longrightarrow\ Ker(\rho)\ \longrightarrow\
\frac{\theta(h)}{\overline{\omega}h(\theta(p))}\
\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}\ \frac{\theta(h)}{T}\
\longrightarrow\ 0$

that $cod\
T=dim_{\mathbb{C}}\frac{\theta(h)}{\overline{\omega}h(\theta(p))}-dim_{\mathbb{C}}Ker(\rho)$.
But $\displaystyle\frac{\theta(h)}{\overline{\omega}h(\theta(p))}$ is
isomorphic to $\bigoplus_{j=1}^{p-k}\frac{{\cal O}_{n}}{f^{*}({\cal O}_{p})}.$

It is well known that the invariants $\delta_{f}$ and $d(f)$ coincide for
parameterized plane curves, but this does not hold in general. In [13,
Proposition 2.9] we proved that parameterized surfaces
$f(x,y)=(x,x^{\lambda}y,y^{m_{1}},y^{m_{2}})$ satisfy $d(f)\leq\delta_{f}$
with equality if and only if $m_{1}=2$.

We prove in the next proposition that these two invariants coincide for any
$f\in{\cal O}(n,2n)$ which has a stable projection equivalent to a fold.
Moreover we obtain an estimate for $Ker(\rho)$ in Theorem 5.5 and consequently
for the ${\mathcal{A}}_{e}$-codimension. For what follows, we denote the ring
${\cal O}_{n}$ by ${\cal O}_{\underline{x},y}$.

######  Proposition 5.6.

Let $f:(\mathbb{C}^{n},0)\to(\mathbb{C}^{p},0)$ be an ${\mathcal{A}}$-finite
map-germ for $p\geq 2n$. We suppose $f\underset{{\mathcal{A}}}{\sim}(g_{n},h)$
such that $g_{n}(\underline{x},y)=(\underline{x},y^{2})$. Thus

  1. 1.

if $p=2n$, then $d(f)=\delta_{f}$;

  2. 2.

if $p\geq 2n$, then $(p-2n+1)\delta_{f}\leq{\cal
A}_{e}cod(f)\leq(p-n)\delta_{f}-1$.

Proof: We can consider change of coordinates such that
$f\underset{{\mathcal{A}}}{\sim}(\underline{x},y^{2},yu_{n+1}(\underline{x},y^{2}),\ldots,yu_{p}(\underline{x},y^{2}))$
for some $u_{n+1},\ldots,u_{p}\in\mathbb{C}\\{\underline{x},y\\}$, as in [10].

If $p=2n$, then by (8) we have that $\varphi_{n}(\underline{x},y,z)=z+y$ and
$\varphi_{j}(\underline{x},y,z)=\frac{zu_{j}(\underline{x},z^{2})-yu_{j}(\underline{x},y^{2})}{z-y}$
for all $j=n+1,\dots,2n$. Identifying $z=-y$ we have
$\varphi_{j}(\underline{x},y,z)=u_{j}(\underline{x},y^{2})$. However ${\cal
O}_{\underline{x},y}\simeq{\cal O}_{\underline{x},y^{2}}\oplus y{\cal
O}_{\underline{x},y^{2}}$ thus

$\langle u_{n+1},\ldots,u_{2n}\rangle{\cal O}_{\underline{x},y}=\langle
u_{n+1},\ldots,u_{2n}\rangle{\cal O}_{\underline{x},y^{2}}\oplus\langle
u_{n+1},\ldots,u_{2n}\rangle\ y{\cal O}_{\underline{x},y^{2}}.$

Hence, $d(f)=\displaystyle\frac{1}{2}dim_{\mathbb{C}}\displaystyle\frac{{\cal
O}_{\underline{x},y}}{\langle u_{n+1},\ldots,u_{2n}\rangle{\cal
O}_{\underline{x},y}}=dim_{\mathbb{C}}\displaystyle\frac{{\cal
O}_{\underline{x},y^{2}}}{\langle u_{n+1},\ldots,u_{2n}\rangle{\cal
O}_{\underline{x},y^{2}}}.$

On the other hand, we can identify $f^{*}({\cal O}_{2n})$ with ${\cal
O}_{\underline{x},y^{2}}\oplus\langle u_{n+1},\ldots,u_{2n}\rangle y{\cal
O}_{\underline{x},y^{2}}$ which implies

$\delta_{f}=\displaystyle dim_{\mathbb{C}}\frac{{\cal
O}_{\underline{x},y^{2}}\oplus y{\cal O}_{\underline{x},y^{2}}}{{\cal
O}_{\underline{x},y^{2}}\oplus\langle u_{n+1},\ldots,u_{2n}\rangle\ y{\cal
O}_{\underline{x},y^{2}}}=dim_{\mathbb{C}}\frac{{\cal
O}_{\underline{x},y^{2}}}{\langle u_{n+1},\ldots,u_{2n}\rangle{\cal
O}_{\underline{x},y^{2}}}=d(f).$

In order to prove the item $2$, notice that it follows from Theorem 5.5 that
${\cal A}_{e}cod(f)=cod\ T$, since $g_{n}$ is a stable map. But
$\omega=(\omega_{n+1},\dots,\omega_{p})\in T$ if and only if $(0,\omega)\in
T{\cal A}_{e}(f)$, that is, there exist
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{n})\in\theta(n)$ and
$\eta=(\eta_{1},\ldots,\eta_{p})\in\theta(p)$ such that
$\epsilon_{i}=-\eta_{i}(f)$ for $i=1,\ldots,n-1$ and
$2y\epsilon_{n}=-\eta_{n}(f)$ which implies that for each $j$ with $n+1\leq
j\leq p$,

$\omega_{j}=-\sum_{i=1}^{n-1}\left(y\frac{\partial u_{j}}{\partial
x_{i}}({\underline{x}},y^{2})\eta_{i}(f)\right)+\left(u_{j}(\underline{x},y^{2})+2y^{2}\frac{\partial
u_{j}}{\partial
y}({\underline{x}},y^{2})\right)\epsilon_{n}(\underline{x},y)+\eta_{j}(f)$

satisfies
$\gamma_{j}:=u_{j}({\underline{x}},y^{2})\epsilon_{n}({\underline{x}},y)-y\frac{\partial
u_{j}}{\partial y}({\underline{x}},y^{2})\eta_{n}(f)+\eta_{j}(f)\in
f^{*}({\cal O}_{p})$. In fact, $y\frac{\partial u_{j}}{\partial
y}({\underline{x}},y^{2})$ is even in $y$ and by isomorphism ${\cal
O}_{\underline{x},y}\simeq{\cal O}_{\underline{x},y^{2}}\oplus y{\cal
O}_{\underline{x},y^{2}}$ we can write
$\epsilon_{n}(\underline{x},y)=\epsilon_{n_{1}}(\underline{x},y^{2})+y\epsilon_{n_{2}}(\underline{x},y^{2})$.
Thus

$u_{j}({\underline{x}},y^{2})\epsilon_{n}({\underline{x}},y)=u_{j}({\underline{x}},y^{2})\epsilon_{n_{1}}(\underline{x},y^{2})+yu_{j}({\underline{x}},y^{2})\epsilon_{n_{2}}(\underline{x},y^{2})\in
f^{*}({\cal O}_{p})$

because $u_{j},\epsilon_{n_{1}},\epsilon_{n_{2}}$ are even in $y$ and
$yu_{j}({\underline{x}},y^{2})\in f^{*}({\cal O}_{p})$. From (10) and the
proof of Theorem 5.5, it follows that
$\omega\in\frac{T}{\overline{\omega}h(\theta(p))}$ if and only if $\omega\in
M$, where

$M:=\frac{f^{*}({\cal
O}_{p})\\{v_{1},\dots,v_{n-1}\\}+\overline{\omega}h(\theta(p))}{\overline{\omega}h(\theta(p))}$

is a $f^{*}({\cal O}_{p})$-module finitely generated by

$v_{i}=\left(\begin{array}[]{c}y\frac{\partial u_{n+1}}{\partial x_{i}}\\\
\vdots\\\ y\frac{\partial u_{p}}{\partial x_{i}}\end{array}\right),$

with $i=1,\ldots,n-1$. By abuse of notation we denote $v_{i}$ its class in
$\overline{\omega}h(\theta(p))$.

Notice that $v_{1},\dots,v_{n-1}$ are not all identically zero in $M$. In
fact, since $p\geq 2n$ and $f$ is ${\mathcal{A}}$ (and therefore ${\cal L}$)
finitely determined, for each $i=1,\ldots,n-1$ there exists
$k_{i}\in\mathbb{N}$ such that $x_{i}^{\gamma_{i}}y\in f^{*}({\cal O}_{p})$
for all $\gamma_{i}\geq k_{i}$. This implies that writing
$u_{n+i}(\underline{x},y^{2})=\phi_{n+i}(\underline{x})+y^{2}R_{n+i}(\underline{x},y)$
there are at least $n-1$ maps $\phi_{j}$’s not identically zero, that we can
suppose they are $\phi_{n+1},\ldots,\phi_{2n-1}$. For each $i=1,\ldots,n-1$
there exists $l_{i}$ such that
$\phi_{l_{i}}(\underline{x})=a_{l_{i}}x_{i}^{l_{i}}+\ldots$ with
$ord_{x_{i}}\phi_{l_{i}}(\underline{x})\leq
ord_{x_{i}}\phi_{j}(\underline{x})$ for all $j$ with
$l_{i},j\in\\{n+1,\ldots,2n-1\\}$. Thus

$ord_{x_{i}}\frac{y\partial u_{l_{i}}(x,y^{2})}{\partial x_{i}}<l_{i}$

and, therefore, the $l_{i}$-coordinate of $v_{i}$ not belongs to $f^{*}({\cal
O}_{p})$, by minimality of the order of $\phi_{l_{i}}$ with respect to
$x_{i}$. Consequently $v_{i}\not\in\overline{\omega}h(\theta(p))$ and we
conclude that $dim_{\mathbb{C}}\ Ker(\rho)\geq 1$.

If $v_{1},\ldots,v_{n-1}$ are non zero in $M$, this means that each non-zero
coordinate of $v_{i}$ is an element of $\frac{{\cal O}_{n}}{f^{*}({\cal
O}_{p})}\simeq\mathbb{C}\\{\lambda_{1},\ldots,\lambda_{\delta_{f}}\\}$. Since
$f$ is ${\cal L}$-finite, for each $i=1,\ldots,n-1$ there exist (finite)
$m_{i1},\ldots,m_{ir_{i}}\in f^{*}({\cal O}_{p})$ such that
$m_{ij}v_{i}\not\in\overline{\omega}h(\theta(p))$, with $1\leq j\leq r_{i}$.
But, in each non zero component of $v_{i},m_{i1}v_{i},\ldots,m_{ir_{i}}v_{i}$
we have at most $\delta_{f}$ linearly independent elements in $\frac{{\cal
O}_{n}}{f^{*}({\cal O}_{p})}$. According to these arguments we have
$dim_{\mathbb{C}}\ M\leq(n-1)\delta_{f}$. Therefore $1\leq dim_{\mathbb{C}}\
Ker(\rho)=dim_{\mathbb{C}}\
\frac{T}{\overline{\omega}h(\theta(p))}\leq(n-1)\delta_{f}$. The result
follows from Theorem 5.5.

######  Corollary 5.7.

If $v_{1},\ldots,v_{n-1}$ (as defined in the above proposition) are
$f^{*}({\cal O}_{p})$-linearly independent, then ${\cal
A}_{e}cod(f)\leq(p-n)\delta_{f}-n+1$.

Proof: If $v_{1},\ldots,v_{n-1}\in M$ are $f^{*}({\cal O}_{p})$-linearly
independent we have $dim_{\mathbb{C}}\ Ker(\rho)\geq n-1$.

######  Example 5.8.

In this example we present ${\mathcal{A}}$-simple map-germs from the
classification obtained in [8]. Notice that the estimates for the
${\mathcal{A}}_{e}$-codimension in Proposition 5.6 are sharp. In fact, the
germs $f(x,y)=(x,xy,y^{2},y^{2k+1})$ and $g(x,y)=(x,y^{2},y^{3},x^{k}y)$
satisfy $\delta_{f}=k=\delta_{g}$. Thus ${\cal A}_{e}cod(f)=k=\delta_{f}$ and
${\mathcal{A}}_{e}cod(g)=2k-1=2\delta_{g}-1$.

Now, the map
$f(x_{1},x_{2},y)=(x_{1},x_{2},y^{2},x_{1}x_{2}y,(x_{1}^{2}-x_{2}^{2})y,y^{3})$
illustrates the computation of $Ker(\rho)$. It is possible to prove that
$\frac{{\cal O}_{3}}{f^{*}({\cal
O}_{6})}\simeq\mathbb{C}\\{y,x_{1}y,x_{2}y,x_{1}^{2}y\\}$, thus
$\delta_{f}=4$. In this case,

$Ker(\rho)\simeq\displaystyle\frac{\mathbb{C}\\{v_{1},v_{2},x_{1}v_{1},x_{1}v_{2}\\}+\overline{\omega}h(\theta(6))}{\overline{\omega}h(\theta(6))}$

is a $f^{*}({\cal O}_{6})$-module generated by $\\{v_{1},v_{2}\\}$, where

$v_{1}=\left(\begin{array}[]{c}x_{1}y\\\ -2x_{2}y\\\ 0\end{array}\right)\ \
\mbox{and}\ \ v_{2}=\left(\begin{array}[]{c}x_{2}y\\\ 2x_{1}y\\\
0\end{array}\right).$

By Theorem 5.5, we conclude that ${\cal
A}_{e}cod(f)=3\delta_{f}-dim_{\mathbb{C}}Ker(\rho)=12-4=8<3\delta_{f}-n+1.$

## References

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Rodrigues Hernandes, M. E.
<EMAIL_ADDRESS>
Universidade Estadual de Maringá, DMA, Av. Colombo 5790. Maringá-PR 87020-900,
Brazil.

Ruas, M. A. S.
<EMAIL_ADDRESS>
Universidade de São Paulo, ICMC, Caixa Postal 668, São Carlos-SP, 13560-970,
Brazil.