$\mathbb{C}$-ana.: there is a unique way to extend a local system on a
punctured small disk at the origin of $\mathbb{C}$ to a local system on
$\mathbb{C}^{\times}$.

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Department of Mathematics, University of California, Berkeley,

Email<EMAIL_ADDRESS>


# New analytical and geometrical aspects of the algebraic multiplicity

Julián López-Gómez, Juan Carlos Sampedro  Institute of Interdisciplinary
Mathematics (IMI)
Department of Analysis and Applied Mathematics
Complutense University of Madrid
28040-Madrid
Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper reveals some new analytical and geometrical properties of the
generalized algebraic multiplicity, $\chi$, introduced in [7, 5] and further
developed in [20, 23, 24]. In particular, it establishes a completely new
connection between $\chi$ and the concept of local intersection index of
algebraic varieties, a central device in Algebraic Geometry. This link between
Nonlinear Spectral Theory and Algebraic Geometry provides to $\chi$ with a
deep geometrical meaning. Moreover, $\chi$ is characterized through the new
notion of local determinant of the Schur operator associated to the linear
path, $\mathfrak{L}(\lambda)$.

###### Key words and phrases:

Algebraic multiplicity, Nonlinear Spectral Theory, Schur complement, local
determinant, bifurcation equation, local intersection index, Algebraic
Geometry

###### 2010 Mathematics Subject Classification:

47J10, 47J15, 58C40, 14C17

The first author has been supported by the Research Grant PGC2018-097104-B-100
of the Spanish Ministry of Science, Technology and Universities and by the
Institute of Interdisciplinar Mathematics of Complutense University. The
second author has been supported by PhD Grant PRE2019_1_0220 of the Basque
Country Government.

##  1\. Introduction

This paper establishes some new properties of the generalized algebraic
multiplicity introduced in [7, 5] and further developed in [20, 23, 24],
denoted by $\chi$ in this paper. In particular, it reveals a hidden connection
between $\chi$ and the concept of local intersection index of algebraic
varieties, a central device in Algebraic Geometry. $\chi$ is a concept of
multiplicity for general curves of linear Fredholm operators,
$\mathfrak{L}(\lambda)$, which has shown to be pivotal in Bifurcation Theory
and Nonlinear Analysis.

Classical Spectral Theory deals with special curves
$\mathfrak{L}\in\mathcal{C}(\Omega,\mathcal{L}_{c}(U,V))$ of the form

$\mathfrak{L}(\lambda)=\lambda I_{U}-K,\qquad\lambda\in\Omega,$

where $U$ and $V$ are two $\mathbb{K}$-Banach spaces such that $U\subset V$,
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$, $\Omega$ is an open domain of
$\mathbb{K}$, $\mathcal{L}_{c}(U,V)$ stands for the space of linear continuous
operators that are compact perturbations of the identity map, $I_{U}$, and $K$
is a given compact operator. Adopting a geometrical perspective, Classical
Spectral Theory studies the intersections of the straight line
$\mathfrak{L}(\lambda)$ with the space of singular operators

$\mathcal{S}(U,V):=\mathcal{L}(U,V)\backslash GL(U,V),$

where $GL(U,V)$ denotes the space of invertible operators. In this context,
$\lambda_{0}\in\Omega$ is said to be an eigenvalue of the straight line
$\mathfrak{L}(\lambda)=\lambda I_{U}-K$ if
$\mathfrak{L}(\lambda_{0})\in\mathcal{S}(U,V)$, i.e., if
$\lambda_{0}\in\sigma(K)$. Note that, in particular, these linear paths lie in
the set of Fredholm operators of index zero, denoted in this paper by
$\Phi_{0}(U,V)$.

More generally, Nonlinear Spectral Theory deals with general continuous paths
in $\Phi_{0}(U,V)$, $\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$,
generalizing the classical theory not only because it deals with arbitrary
continuous curves, not necessarily straight lines, but also because these
paths can lie in $\Phi_{0}(U,V)$ remaining outside
$\mathcal{L}_{c}(U,V)\varsubsetneq\Phi_{0}(U,V)$. In this general context,
given a Fredholm path, $\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$, a
value $\lambda\in\Omega$ is said to be a _generalized eigenvalue_ of
$\mathfrak{L}$ if $\mathfrak{L}(\lambda)\notin GL(U,V)$. Then, the
_generalized spectrum_ of $\mathfrak{L}$, $\Sigma(\mathfrak{L})$, consists of
the set of generalized eigenvalues, i.e.,

$\Sigma(\mathfrak{L}):=\\{\lambda\in\Omega:\mathfrak{L}(\lambda)\notin
GL(U,V)\\}.$

Thus, as in the classical case, $\Sigma(\mathfrak{L})$ also consists of the
intersection points of the curve $\mathfrak{L}$ with the singular manifold
$\mathcal{S}(U,V)\cap\Phi_{0}(U,V)$. Naturally, for every compact operator
$K$,

$\Sigma(\lambda I_{U}-K)=\sigma(K).$

An important ingredient in Nonlinear Spectral Theory is the concept of
generalized algebraic multiplicity, which assigns to every pair
$(\mathfrak{L},\lambda_{0})$ with
$\mathfrak{L}\in\mathcal{C}^{\infty}(\Omega,\Phi_{0}(U,V))$ and
$\lambda_{0}\in\Sigma(\mathfrak{L})$, an integer
$\chi[\mathfrak{L},\lambda_{0}]\in[1,+\infty]$ such that, whenever $\dim
U=\dim V<\infty$,

(1.1)
$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\det\mathfrak{L}(\lambda).$

Although a number of constructions of $\chi[\mathfrak{L},\lambda_{0}]$ had
been done by Göhberg and Sigal [11], Magnus [25], Ize [15], Esquinas and
López-Gómez [7], Esquinas [6] and Rabier [28], it was not until 2001, that
Chapters 4 and 5 of López-Gómez [20] characterized whether all these
generalized algebraic multiplicities were finite through the pivotal concept
of _algebraic eigenvalue_. A generalized eigenvalue, $\lambda_{0}\in\Omega$,
of a continuous path of Fredholm operators,
$\mathfrak{L}(\lambda)\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$, is said to be
$\kappa$-algebraic if there exist $\varepsilon>0$ and $C>0$ such that
$\mathfrak{L}(\lambda)$ is an isomorphism whenever
$0<|\lambda-\lambda_{0}|<\varepsilon$, and

(1.2)
$\|\mathfrak{L}^{-1}(\lambda)\|<\frac{C}{|\lambda-\lambda_{0}|^{\kappa}}\quad\hbox{if}\;\;0<|\lambda-\lambda_{0}|<\varepsilon,$

with $\kappa$ the minimal integer for which (1.2) holds. Throughout this
paper, the set of $\kappa$-transversal eigenvalues of $\mathfrak{L}(\lambda)$
is denoted by $\operatorname{Alg}_{\kappa}(\mathfrak{L})$, and the set of
_algebraic eigenvalues_ of $\mathfrak{L}$ is defined by

$\operatorname{Alg}(\mathfrak{L}):=\bigcup_{\kappa\in\mathbb{N}}\operatorname{Alg}_{\kappa}(\mathfrak{L}).$

It turns out that the multiplicities of [25, 7, 6] are well defined if, and
only if, the path $\mathfrak{L}(\lambda)$ is of class $\mathcal{C}^{r}$ for
some $r\geq 1$ and $\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathfrak{L})$
for some $1\leq\kappa\leq r$. Moreover, by Theorems 4.4.1 and 4.4.4 of López-
Gómez [20], when $\mathfrak{L}$ is analytic and $\mathfrak{L}(\Omega)$
contains some invertible operator, then $\Sigma(\mathfrak{L})$ is discrete,
and every $\lambda_{0}\in\Sigma(\mathfrak{L})$ is an algebraic eigenvalue of
$\mathfrak{L}$. Thus, through this paper, it is convenient to denote by
$\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$ the set of
curves $\mathfrak{L}\in\mathcal{C}^{r}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$
such that
$\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathfrak{L})\cap\Omega_{\lambda_{0}}$
with $1\leq\kappa\leq r$ for some $r\in\mathbb{N}$. According to [20, Ch. 4],
the generalized algebraic multiplicity $\chi[\mathfrak{L},\lambda_{0}]$ is
well defined if and only if
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$.

Short time later, in 2004, the theory of algebraic multiplicities for
$\mathcal{C}^{\infty}$-Fredholm paths was axiomatized by Mora-Corral [26] by
establishing that, modulus a normalization condition (see Theorem 2.2 for the
precise statement), given an open connected neighborhood of $\lambda_{0}$ in
$\mathbb{K}$, denoted by $\Omega_{\lambda_{0}}$, the algebraic multiplicity
$\chi$ is the unique map

$\chi[\cdot,\lambda_{0}]:\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(U))\longrightarrow[0,\infty]$

satisfying the _product formula_

$\chi[\mathfrak{L}\circ\mathfrak{M},\lambda_{0}]=\chi[\mathfrak{L},\lambda_{0}]+\chi[\mathfrak{M},\lambda_{0}]$

for all
$\mathfrak{L},\mathfrak{M}\in\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(U))$,
regardless whether, or not, $\lambda_{0}$ is singular. These findings were
collected by López-Gómez and Mora-Corral in [23], where, in addition, the
theory of Göhberg and Sigal [11] was substantially generalized to a non-
holomorphic setting, and the existence of the local Smith form was established
in the class $\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$
through the lengths of the Jordan chains of $\mathfrak{L}(\lambda)$.

The relevance of $\chi[\mathfrak{L},\lambda_{0}]$ in Nonlinear Analysis relies
on the crucial fact that, in the special case when $U=V$,
$\mathbb{K}=\mathbb{R}$, $\Omega_{\lambda_{0}}=(a,b)$ for some $a<b$,
$\lambda_{0}\in(a,b)$ is isolated in $\Sigma(\mathfrak{L})$, and
$I_{U}-\mathfrak{L}(\lambda)$ is compact for all $\lambda\in(a,b)$, by [20,
Th. 5.6.2], $\chi[\mathfrak{L},\lambda_{0}]$ is odd if and only if the
Leray–Schauder degree $\deg_{LS}(\mathfrak{L}(\lambda),B_{R}(0))$ changes as
$\lambda$ crosses $\lambda_{0}$ (see López-Gómez and Sampedro [24] for far
more general results in the Fredholm context). In this paper, for any given
$u_{0}\in U$ and $R>0$, we denote by $B_{R}(u_{0})$ the ball of radius $R$
centered at $u_{0}$. Note that, whenever $\dim U<+\infty$, one can chose a
basis in $U$ so that

$\deg_{LS}(\mathfrak{L}(\lambda),B_{R}(0))=\mathrm{sign\,det}\mathfrak{L}(\lambda)$

for all $\lambda\in(a,b)\setminus\Sigma(\mathfrak{L})$. These changes of
degree have relevant consequences in Nonlinear Analysis, because they entail
that, for every completely continuous map $\mathfrak{N}(\lambda,u)$ such that
$\mathfrak{N}(\lambda,0)=0$ and $\mathfrak{N}(\lambda,u)=o(\|u\|)$ as $u\to
0$, the set of nontrivial solutions of the equation

(1.3)
$\mathscr{F}(\lambda,u)\equiv\mathfrak{L}(\lambda)u+\mathfrak{N}(\lambda,u)=0$

possesses a connected component, denoted by $\mathscr{C}$, bifurcating from
$u=0$ at $\lambda=\lambda_{0}$ (see [20, 21]). By a nontrivial solution it is
meant any pair $(\lambda,u)\in\mathscr{F}^{-1}(0)$ with $u\neq 0$.
Surprisingly, this occurs for every completely continuous nonlinearity
$\mathfrak{N}(\lambda,u)$ if and only if $\chi[\mathfrak{L},\lambda_{0}]$ is
odd. And, in such case, the component $\mathscr{C}$ satisfies the _global
alternative_ of Rabinowitz [29], as well as the global theorems of Nirenberg
[27] and Magnus [25] (see [20, Th. 6.3.1]). Indeed, when $(a,b)=\mathbb{R}$,
either

  1. (a)

$\mathscr{C}$ is unbounded in $\mathbb{R}\times U$; or

  2. (b)

there exists $\lambda_{1}\neq\lambda_{0}$ such that
$(\lambda_{1},0)\in\overline{\mathscr{C}}$.

The Leray–Schauder degree had been incorporated to the weaponry of Nonlinear
Integral Equations by Krasnoselskij [16] in order to prove his celebrated
local bifurcation theorem, establishing that if $K:U\to U$ is compact,
$\mathfrak{L}(\lambda)=\lambda I_{U}-K$ for all $\lambda\in(a,b)$, and
$\lambda_{0}\in\sigma(K)$ is an eigenvalue with odd (classical) algebraic
multiplicity, then the component $\mathscr{C}$ exists for every completely
continuous map $\mathfrak{N}(\lambda,u)$ satisfying the previous requirements.

The first aim of this paper is to identify $\chi[\mathfrak{L},\lambda_{0}]$ in
terms of the _Schur operator_ of the path $\mathfrak{L}(\lambda)$. This
provides us with a definition of $\chi$ as a natural extension of the finite
dimensional concept (1.1) adopting a direct approach. In this paper, for any
given $T\in\Phi_{0}(U,V)$, we will denote by $N[T]$ the null space, or kernel,
of $T$, and by $R[T]$ the range, or image, of $T$. Assume
$\lambda_{0}\in\Sigma(\mathfrak{L})$ and let $P\in\mathcal{L}(U)$ and
$Q\in\mathcal{L}(V)$ be a pair of projections onto
$N[\mathfrak{L}(\lambda_{0})]$ and $R[\mathfrak{L}(\lambda_{0})]$,
respectively. Then,

(1.4) $U=(I_{U}-P)(U)\oplus N[\mathfrak{L}(\lambda_{0})],\qquad
V=R[\mathfrak{L}(\lambda_{0})]\oplus(I_{V}-Q)(V),$

and $\mathfrak{L}(\lambda)$ can be expressed as a block operator matrix

(1.5)
$\mathfrak{L}(\lambda)=\left(\begin{array}[]{cc}L_{11}(\lambda)&L_{12}(\lambda)\\\\[4.30554pt]
L_{21}(\lambda)&L_{22}(\lambda)\end{array}\right)$

where

(1.6)
$\begin{array}[]{ll}L_{11}(\lambda):=Q\mathfrak{L}(\lambda)(I_{U}-P),&\quad
L_{12}(\lambda):=Q\mathfrak{L}(\lambda)P,\\\
L_{21}(\lambda):=(I_{V}-Q)\mathfrak{L}(\lambda)(I_{U}-P),&\quad
L_{22}(\lambda):=(I_{V}-Q)\mathfrak{L}(\lambda)P.\end{array}$

In this context, the Schur operator of $\mathfrak{L}(\lambda)$ associated to
the projection pair $(P,Q)$ can be defined through

$\mathscr{S}_{\mathfrak{L}(\lambda),(P,Q)}(\mathfrak{L}(\lambda)):=L_{22}(\lambda)-L_{21}(\lambda)L_{11}^{-1}(\lambda)L_{12}(\lambda),\qquad\lambda\in\Omega_{\lambda_{0}}.$

In terms of this operator, our first result can be stated as follows.

######  Theorem 1.1.

Assume $\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$ and
$\lambda_{0}\in\Omega\cap\Sigma(\mathfrak{L})$ is an isolated eigenvalue.
Then, for every pair $(P,Q)$ of $\mathfrak{L}(\lambda_{0})$-projections,

(1.7)
$\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=P\mathfrak{L}^{-1}(\lambda)(I_{V}-Q).$

If, in addition,
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$,
then

(1.8)
$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\det\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\operatorname{ord}_{\lambda_{0}}\det[P\mathfrak{L}^{-1}(\lambda)(I_{V}-Q)]^{-1}.$

Theorem 1.1 allows us to introduce, in a rather natural manner, a local
concept of infinite-dimensional determinant,
$\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}$, for which

$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\operatorname{ord}_{\lambda_{0}}\det\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda)).$

Indeed, we will see that taking

$\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda)):=\det(L_{11}(\lambda))\cdot\det(\mathscr{S}_{T,(P,Q)}(\mathfrak{L}(\lambda))$

is consistent with the classical theory of Schur complements. As a byproduct
of Theorem 1.1, the multiplicity of Ize [15], introduced for analytic paths,
can be defined _mutatis mutandis_ for every admissible path
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$.
Our next result establishes a sharp connection between these concepts and
Bifurcation Theory.

######  Theorem 1.2.

Assume $\mathfrak{L}\in\mathcal{C}^{1}(\Omega,\Phi_{0}(U,V))$ and
$\lambda_{0}\in\Omega$. Then, for every pair $(P,Q)$ of
$\mathfrak{L}(\lambda_{0})$-projections, the Schur operator
$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))$
coincides with the linearization at $(\lambda,u)=(\lambda,0)$ of the
bifurcation equation of (1.3) through the Lyapunov–Schmidt decomposition of
projections $(P,Q)$, derived as discussed in [20, Ch. 3], which is denoted by
$\mathscr{B}(\lambda)$ in this paper (see Section 4), i.e.

(1.9)
$\mathscr{B}(\lambda)=\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=L_{22}(\lambda)-L_{21}(\lambda)L_{11}^{-1}(\lambda)L_{12}(\lambda).$

If, in addition,
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$,
then, thanks to Theorem 1.1,

(1.10)
$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\det\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\operatorname{ord}_{\lambda_{0}}\det\mathscr{B}(\lambda),$

regardless the pair of projections chosen, $(P,Q)$. Therefore, thanks to
Theorem [20, Th. 4.3.4], the following assertions are equivalent:

  1. (a)

$\lambda_{0}$ is a nonlinear eigenvalue of $\mathfrak{L}(\lambda)$, in the
sense that $(\lambda_{0},0)$ is a bifurcation point of (1.3) from
$(\lambda,u)=(\lambda,0)$ for all map $\mathfrak{N}(\lambda,u)$ of class
$\mathcal{C}^{1}$.

  2. (b)

$\chi[\mathfrak{L},\lambda_{0}]$ is an odd integer.

  3. (c)

$\det\mathscr{B}(\lambda)$ changes of sign as $\lambda$ crosses $\lambda_{0}$,
regardless $(P,Q)$.

  4. (d)

$\det\mathscr{S}_{\mathfrak{L}(\lambda),(P,Q)}(\mathfrak{L}(\lambda))$ changes
of sign as $\lambda$ crosses $\lambda_{0}$, regardless $(P,Q)$.

  5. (e)

$\det[P\mathfrak{L}^{-1}(\lambda)(I_{V}-Q)]$ changes of sign as $\lambda$
crosses $\lambda_{0}$, regardless $(P,Q)$.

Furthermore, when any of these conditions occurs, there is a component,
$\mathscr{C}$, of the set of nontrivial solutions of (4.1) such that
$(\lambda_{0},0)\in\overline{\mathscr{C}}$. By a component, it is meant any
closed and connected subset maximal for the inclusion.

Our second aim is incorporating a completely new perspective into the theory
of algebraic multiplicities in order to establish a bisociation between
Spectral Theory and Algebraic Geometry. As already discussed above, any
generalized eigenvalue $\lambda_{0}\in\Sigma(\mathfrak{L})$ of the curve
$\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$ lies in the intersection of
$\mathfrak{L}$ with the singular space $\mathcal{S}(U,V)\cap\Phi_{0}(U,V)$.
This geometrical feature leads us to think that the algebraic multiplicity
might actually measure how the curve $\mathfrak{L}(\lambda)$ intersects
geometrically with $\mathcal{S}(U,V)\cap\Phi_{0}(U,V)$. Thus, it might be
related to the local intersection index, a pivotal geometrical device for
measuring the nature of the intersections of varieties.

Given a family of algebraic varieties, $\mathscr{V}$, an intersection theory
over $\mathscr{V}$ consists of giving a pairing

(1.11) $\bullet\,:\;A^{r}(X)\times A^{s}(X)\to A^{r+s}(X)$

satisfying a series of axioms (see, e.g., Hartshorne [14, pp. 426–427],
Eisenbud and Harris [5, Ch. 1 & 2], and Fulton [13, Ch. 7 & 8]) for every
$r,s\in\mathbb{N}$ and $X\in\mathscr{V}$, where $A^{r}(X)$ stands for the
group of cycles of codimension $r$ on $X$ modulo rational equivalence. The
graded group

$A(X)\equiv\bigoplus_{r\in\mathbb{N}}A^{r}(X)$

is referred to as the Chow group after Chow [3]. The pairing (1.11) gives to
$A(X)$ the structure of a graded ring, the Chow ring of $X$. Giving an
intersection theory to an algebraic variety $X$ consists in giving the
structure of the Chow ring $A(X)$; the axioms of the pairing $\bullet$ try to
mimic in $X$ the celebrated Bezout Theorem [5, & 2.1.1]. This explains why one
of these axioms establishes that if $X_{1}$ and $X_{2}$ are subvarieties of
$X$ with _proper intersection_ , in the sense that any irreducible component
of $X_{1}\cap X_{2}$ has codimension
$\operatorname{codim}X_{1}+\operatorname{codim}X_{2}$, then

$[X_{1}]\bullet[X_{2}]=\sum_{j}i(X_{1},X_{2};C_{j})[C_{j}]$

where the sum runs over the set of all irreducible components, $C_{j}$, of
$X_{1}\cap X_{2}$, and the integer $i(X_{1},X_{2};C_{j})$ stands for the local
intersection index of $X_{1}$ and $X_{2}$ along $C_{j}$. By a result of Serre
[30, Ch. V, & C.1], for any given pair $X_{1},X_{2}$ of subvarieties of a
smooth variety $X$ with proper intersection and any irreducible component,
$C$, of $X_{1}\cap X_{2}$, the local intersection index of $X_{1}$ and $X_{2}$
along $C$ is given through

$i(X_{1},X_{2};C)=\sum_{i=0}^{\dim
X}(-1)^{i}\ell_{\mathcal{O}_{c,X}}\operatorname{Tor}^{\mathcal{O}_{c,X}}_{i}(\mathcal{O}_{c,X}/\mathfrak{P}_{X_{1}},\mathcal{O}_{c,X}/\mathfrak{P}_{X_{2}})$

where $\mathcal{O}_{c,X}$ is the local ring of $c\in C$ in $X$, and
$\mathfrak{P}_{X_{1}}$ and $\mathfrak{P}_{X_{2}}$ are the ideals of $X_{1}$
and $X_{2}$, respectively, in the ring $\mathcal{O}_{c,X}$. In this paper,
and, in particular, in the Serre formula, we are denoting by $\ell_{R}(M)$ the
length of the module $M$ over the ring $R$, and by
$\operatorname{Tor}_{i}^{R}$ the $i$-th Tor $R$-module. As in our context $X$
is a smooth algebraic variety over an algebraically closed field,
$\mathbb{C}$, and $X_{1},X_{2}$ are two irreducible Cohen–Macaulay
subvarieties of $X$ such that $X_{1}\cap X_{2}=\\{x\\}$, with proper
intersection, according to, e.g., Eisenbaud and Harris [5, p. 48], the local
intersection index of $X_{1}$ and $X_{2}$ at $x$ reduces to

(1.12)
$i(X_{1},X_{2};x):=\ell_{\mathcal{O}_{x,X}}(\mathcal{O}_{x,X}/(\mathfrak{P}_{X_{1}}+\mathfrak{P}_{X_{2}})).$

The third finding of this paper establishes a connection between the concepts
of classical algebraic multiplicity and the intersection index. Subsequently,
$\mathcal{D}:\mathcal{L}(\mathbb{C}^{N})\to\mathbb{C}$ stands for the
determinant map defined by

$\mathcal{D}(T):=\det T\quad\hbox{for
every}\;\;T\in\mathcal{L}(\mathbb{C}^{N}).$

######  Theorem 1.3.

Let $T\in\mathcal{L}(\mathbb{C}^{N})$ and $\lambda_{0}\in\sigma(T)$ be, and
set $\mathfrak{L}(\lambda)=\lambda I_{N}-T$, $\lambda\in\mathbb{C}$. Then,

(1.13)
$\chi[\mathfrak{L},\lambda_{0}]=i(\mathcal{D}^{-1}(0),\mathfrak{L}(\mathbb{C});\mathfrak{L}(\lambda_{0})).$

Finally, in order to state our last finding, for any given
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$,
we will denote by

$\mathscr{L}=\mathscr{L}(\mathfrak{L})\in\mathcal{L}(\mathbb{C}^{M})$

the _global linearization_ , as discussed by Lemma 10.1.1 of [23], of the
_local Smith form_ of the Schur complement of $\mathfrak{L}(\lambda)$, whose
existence follows from Theorem 7.4.1 of López-Gómez and Mora-Corral [23]. The
following result is a substantial, very sharp, generalization of Theorem 1.3.

######  Theorem 1.4.

For every
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$,

(1.14) $\chi[\mathfrak{L},\lambda_{0}]=i(\mathcal{D}^{-1}(0),\lambda
I_{M}-\mathscr{L};\lambda_{0}I_{M}-\mathscr{L}).$

This paper is distributed as follows. Section 2 reviews briefly the
generalized algebraic multiplicity $\chi[\mathfrak{L},\lambda_{0}]$. In
Section 3 we introduce a local notion of determinant for a general path
$\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$ through the Schur operator
$\mathscr{S}_{\mathfrak{L}(\lambda),(P,Q)}(\mathfrak{L}(\lambda))$ associated
to a projection pair $(P,Q)$. Then, we use this concept of infinite-
dimensional determinant to prove Theorem 1.1. Section 4 consists of the proof
of Theorem 1.2, Section 5 shows Theorem 1.3 and, finally, Section 6 proves
Theorem 1.4. In reading this paper, it would be appropriate to have copies of
[20] and [23].

##  2\. Nonlinear spectral theory. Generalized algebraic multiplicity.

Throughout this section, $\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$,
where $\Omega$ is a subdomain of $\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$.
According to Theorems 4.4.1 and 4.4.4 of [20], when $\mathfrak{L}(\lambda)$ is
analytic in $\Omega$, i.e.,
$\mathfrak{L}\in\mathcal{H}(\Omega,\Phi_{0}(U,V))$, either
$\Sigma(\mathfrak{L})=\Omega$, or $\Sigma(\mathfrak{L})$ is discrete and
$\Sigma(\mathfrak{L})\subset\operatorname{Alg}(\mathfrak{L})$.

The next concept, going back to Esquinas and López-Gómez [7], is pivotal in
Nonlinear Spectral Theory as it allows to introduce a generalized algebraic
multiplicity, $\chi[\mathfrak{L},\lambda_{0}]$, in a rather natural manner.
Subsequently, we will denote

$\mathfrak{L}_{j}:=\frac{1}{j!}\mathfrak{L}^{(j)}(\lambda_{0}),\quad 0\leq
j\leq r,$

if these derivatives exist.

######  Definition 2.1.

Let $\mathfrak{L}\in\mathcal{C}^{r}(\Omega,\Phi_{0}(U,V))$ and
$1\leq\kappa\leq r$. Then, a given $\lambda_{0}\in\Sigma(\mathfrak{L})$ is
said to be a $\kappa$-transversal eigenvalue of $\mathfrak{L}$ if

$\bigoplus_{j=1}^{\kappa}\mathfrak{L}_{j}\left(\bigcap_{i=0}^{j-1}N[\mathfrak{L}_{i}]\right)\oplus
R(\mathfrak{L}_{0})=V\;\;\hbox{with}\;\;\mathfrak{L}_{\kappa}\left(\bigcap_{i=0}^{\kappa-1}N[\mathfrak{L}_{i}]\right)\neq\\{0\\}.$

For these eigenvalues, the _algebraic multiplicity of $\mathfrak{L}$ at
$\lambda_{0}$_, $\chi[\mathfrak{L},\lambda_{0}]$, was introduced in [7]
through

$\chi[\mathfrak{L},\lambda_{0}]:=\sum_{j=1}^{\kappa}j\cdot\dim\mathfrak{L}_{j}\left(\bigcap_{i=0}^{j-1}N[\mathfrak{L}_{i}]\right).$

According to Theorems 4.3.2 and 5.3.3 of [20], for every
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega,\Phi_{0}(U,V))$ there exists
a polynomial $\Phi:\mathbb{K}\to\mathcal{L}(U)$ with $\Phi(\lambda_{0})=I_{U}$
and an integer $\kappa\in\\{1,2,...,r\\}$ such that $\lambda_{0}$ is a
$\kappa$-transversal eigenvalue of the path

(2.1)
$\mathfrak{L}^{\Phi}:=\mathfrak{L}\circ\Phi\in\mathcal{C}^{r}(\Omega,\Phi_{0}(U,V)).$

Moreover, $\kappa$ and $\chi[\mathfrak{L}^{\Phi},\lambda_{0}]$ are independent
of the curve of _trasversalizing local isomorphisms_ $\Phi$ chosen to
transversalize $\mathfrak{L}$ at $\lambda_{0}$ through (2.1), regardless
$\Phi$ is a polynomial or not. Therefore, the next generalized concept of
algebraic multiplicity, defined for every
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega,\Phi_{0}(U,V))$, is
consistent

$\chi[\mathfrak{L},\lambda_{0}]:=\chi[\mathfrak{L}^{\Phi},\lambda_{0}].$

This concept can be extended by setting

$\chi[\mathfrak{L},\lambda_{0}]:=0\quad\hbox{if}\;\;\lambda_{0}\notin\Sigma(\mathfrak{L})$

and

(2.2)
$\chi[\mathfrak{L},\lambda_{0}]:=+\infty\quad\hbox{if}\;\;\lambda_{0}\in\Sigma(\mathfrak{L})\setminus\operatorname{Alg}(\mathfrak{L})\;\;\hbox{and}\;\;r=+\infty,$

so that $\chi[\mathfrak{L},\lambda]$ is well defined for all
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega,\Phi_{0}(U,V))$ at any
$\lambda\in\Omega$, and, in particular, for every analytic curve
$\mathfrak{L}\in\mathcal{H}(\Omega,\Phi_{0}(U,V))$. Setting (2.2), the
algebraic multiplicity is also well defined for each smooth curve
$\mathfrak{L}\in\mathcal{C}^{\infty}(\Omega,\Phi_{0}(U,V))$ at any
$\lambda\in\Omega$, and it coincides with the multiplicity of Magnus [25]. The
next uniqueness result, going back to Mora-Corral [26], axiomatizes the
concept of Algebraic Multiplicity (see also Chapter 6 of [23]).

######  Theorem 2.2.

For every $\lambda_{0}\in\mathbb{K}$ and any open neighborhood
$\Omega_{\lambda_{0}}\subset\mathbb{K}$ of $\lambda_{0}$, the algebraic
multiplicity $\chi$ is the unique map

$\chi[\cdot,\lambda_{0}]:\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(U))\longrightarrow[0,\infty]$

such that

  1. (PF)

For every pair
$\mathfrak{L},\mathfrak{M}\in\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(U))$,

$\chi[\mathfrak{L}\circ\mathfrak{M},\lambda_{0}]=\chi[\mathfrak{L},\lambda_{0}]+\chi[\mathfrak{M},\lambda_{0}].$

  2. (NP)

There exists a rank one projection $\Pi\in\mathcal{L}(U)$ such that

$\chi[(\lambda-\lambda_{0})\Pi+I_{U}-\Pi,\lambda_{0}]=1.$

The axiom (PF) is the _product formula_ and (NP) is a _normalization property_
for establishing the uniqueness of $\chi$. From these two axioms one can
derive the remaining properties of $\chi$; among them, that it equals the
classical algebraic multiplicity when

(2.3) $\mathfrak{L}(\lambda)=\lambda I_{U}-K$

for some compact operator $K$. Indeed, according to [23], for every
$\mathfrak{L}\in\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(U))$, the
following properties are satisfied:

  * •

$\chi[\mathfrak{L},\lambda_{0}]\in\mathbb{N}\uplus\\{+\infty\\}$;

  * •

$\chi[\mathfrak{L},\lambda_{0}]=0$ if and only if
$\mathfrak{L}(\lambda_{0})\in GL(U)$;

  * •

$\chi[\mathfrak{L},\lambda_{0}]<\infty$ if and only if
$\lambda_{0}\in\operatorname{Alg}(\mathfrak{L})$.

  * •

If $U=\mathbb{K}^{N}$, then, in any basis,

(2.4)
$\chi[\mathfrak{L},\lambda_{0}]=\mathrm{ord}_{\lambda_{0}}\det\mathfrak{L}(\lambda).$

  * •

Let $L\in\mathcal{L}(U)$ be such that $\lambda I_{U}-L\in\Phi_{0}(U)$. Then,
for each eigenvalue $\lambda_{0}\in\sigma(L)$, there exists $\kappa\geq 1$
such that

$\begin{split}\chi[\lambda
I_{U}-L,\lambda_{0}]=\underset{n\in\mathbb{N}}{\sup}\dim
N[(\lambda_{0}I_{U}-L)^{n}]=\dim N[(\lambda_{0}I_{U}-L)^{\kappa}].\end{split}$

  * •

Let $V$ be another complex Banach space and
$\mathfrak{P}\in\mathcal{C}^{\infty}(\Omega_{\lambda_{0}},\Phi_{0}(V))$. Then,

$\chi[\mathfrak{L}\oplus\mathfrak{P},\lambda_{0}]=\chi[\mathfrak{L},\lambda_{0}]+\chi[\mathfrak{P},\lambda_{0}].$

Therefore, $\chi$ extends, very substantially, the classical concept of
algebraic multiplicity, which is only valid for operator paths of the form
(2.3).

##  3\. Schur complement and local determinant

This section gives a new equivalent definition of $\chi$, without invoking the
concept of transversalization, based on the formulation of the finite-
dimensional algebraic multiplicity in terms of the order of the determinant.
To accomplish this task, we need to develop an analogue of the notion of
determinant in an infinite-dimensional setting. Although the notion of
determinant does not exists in such setting, we will see that it can be
introduced in local neighborhoods of $\Phi_{0}(U,V)$.

Throughout this section, $T\in\Phi_{0}(U,V)$ is an arbitrary linear Fredholm
operator of index zero, and $P\in\mathcal{L}(U)$, $Q\in\mathcal{L}(V)$ are
projections onto $N[T]$ and $R[T]$, respectively; The pair $(P,Q)$ will be
called a pair of $T$-projections. Then, the following topological direct sum
decompositions hold

(3.1) $U=(I_{U}-P)(U)\oplus N[T],\qquad V=R[T]\oplus(I_{V}-Q)(V),$

and, setting

$\displaystyle R[I_{U}-P]$ $\displaystyle=(I_{U}-P)(U)\equiv N[T]^{\perp},$
$\displaystyle R[I_{V}-Q]$ $\displaystyle=(I_{V}-Q)(V)\equiv R[T]^{\perp},$

by Fitzpatrick and Pejsachowicz [12, p. 286], for any given
$L\in\Phi_{0}(U,V)$, $L$ can be equivalently expressed as a block operator
matrix

(3.2) $L=\left(\begin{array}[]{cc}L_{11}&L_{12}\\\\[4.30554pt]
L_{21}&L_{22}\end{array}\right)$

where

$\begin{array}[]{ll}L_{11}:=QL(I_{U}-P),&\quad L_{12}:=QLP,\\\
L_{21}:=(I_{V}-Q)L(I_{U}-P),&\quad L_{22}:=(I_{V}-Q)LP.\end{array}$

In particular, since $TP=0$ and $(I_{V}-Q)T=0$, the operator $T$ can be
expressed as

$T=\left(\begin{array}[]{cc}T_{11}&0\\\\[4.30554pt] 0&0\end{array}\right)$

with $T_{11}\in GL(N[T]^{\perp},R[T])$. Since $GL(N[T]^{\perp},R[T])$ is open
in $\mathcal{L}(N[T]^{\perp},R[T])$ and $\Phi_{0}(U,V)$ is open in
$\mathcal{L}(U,V)$, there exists $\varepsilon>0$ such that, whenever
$L\in\mathcal{L}(U,V)$ satisfies $\|T-L\|<\varepsilon$, then
$L\in\Phi_{0}(U,V)$, and it can be expressed as (3.2) with $L_{11}\in
GL(N[T]^{\perp},R[T])$.

In this context, the Schur operator of $T$ associated to the projection pair
$(P,Q)$ can be defined through

$\begin{array}[]{lccl}\mathscr{S}_{T,(P,Q)}:&B_{\varepsilon}(T)\subset\Phi_{0}(U,V)&\longrightarrow&\mathcal{L}(N[T],R[T]^{\perp})\\\
&L&\mapsto&L_{22}-L_{21}L_{11}^{-1}L_{12}\end{array}$

where $B_{\varepsilon}(T)$ stands for the open ball of radius $\varepsilon$
centered at $T\in\Phi_{0}(U,V)$ in $\mathcal{L}(U,V)$. We call this map Schur
operator by analogy, in the Euclidean space, to the Schur complement of a
matrix. Indeed, given any block matrix

(3.3) $M=\left(\begin{array}[]{cc}A&B\\\\[4.30554pt] C&D\end{array}\right)$

with $A\in GL(\mathbb{K}^{n})$, $B\in\operatorname{Mat}_{n\times
m}(\mathbb{K})$, $C\in\operatorname{Mat}_{m\times n}(\mathbb{K})$ and
$D\in\operatorname{Mat}_{m}(\mathbb{K})$, the Schur complement of the matrix
$D$ in $M$ is the matrix $M/A\in\operatorname{Mat}_{m}(\mathbb{K})$ defined by

$M/A:=D-CA^{-1}B.$

By a lemma of Banachiewicz [1, p. 50], the Schur complement satisfies the
identity

$\det(M)=\det(A)\cdot\det(M/A)$

for every block matrix (3.3). Based on this feature, we introduce the
following local notion of determinant.

######  Definition 3.1.

Let $T\in\Phi_{0}(U,V)$ and $(P,Q)$ be a pair of $T$-projections. Then, for
sufficiently small $\varepsilon>0$, we define the local determinant functional
by

$\begin{array}[]{lccl}\mathcal{D}_{T,(P,Q)}:&B_{\varepsilon}(T)\subset\Phi_{0}(U,V)&\longrightarrow&\mathbb{K}\\\
&L&\mapsto&\det(L_{11})\cdot\det(\mathscr{S}_{T,(P,Q)}(L)).\end{array}$

The next result establishes that the local determinant functional indeed
behaves as a (local) determinant on $B_{\varepsilon}(T)$.

######  Theorem 3.2.

Let $T\in\Phi_{0}(U,V)$ and $(P,Q)$ be a pair of $T$-projections. Then, for
sufficiently small $\epsilon>0$ and every $L\in B_{\varepsilon}(T)$,

$L\in GL(U,V)\text{ if and only if }\mathcal{D}_{T,(P,Q)}(L)\neq 0.$

###### Proof.

Let $L\in B_{\varepsilon}(T)$ be. Then, the Schur operator of $L$ is well
defined. Moreover, $L$ can be decomposed as

(3.4) $L=\left(\begin{array}[]{cc}I_{R[T]}&0\\\\[4.30554pt]
L_{21}L^{-1}_{11}&I_{R[T]^{\perp}}\end{array}\right)\left(\begin{array}[]{cc}L_{11}&0\\\\[4.30554pt]
0&\mathscr{S}_{T,(P,Q)}(L)\end{array}\right)\left(\begin{array}[]{cc}I_{N[T]^{\perp}}&L^{-1}_{11}L_{12}\\\\[4.30554pt]
0&I_{N[T]}\end{array}\right).$

Clearly, the first and the third matrix operators of (3.4) are invertible.
Thus, taking inverses, we find that

(3.5)
$\begin{split}L_{11}\oplus\mathscr{S}_{T,(P,Q)}(L)&=\left(\begin{array}[]{cc}L_{11}&0\\\\[4.30554pt]
0&\mathscr{S}_{T,(P,Q)}(L)\end{array}\right)\\\
&=\left(\begin{array}[]{cc}I_{R[T]}&0\\\\[4.30554pt]
L_{21}L^{-1}_{11}&I_{R[T]^{\perp}}\end{array}\right)^{-1}L\left(\begin{array}[]{cc}I_{N[T]^{\perp}}&L^{-1}_{11}L_{12}\\\\[4.30554pt]
0&I_{N[T]}\end{array}\right)^{-1}.\end{split}$

If $L\in GL(U,V)$, then, due to (3.5), $L_{11}\oplus\mathscr{S}_{T,(P,Q)}\in
GL(U,V)$ because it is a composition of invertible operators. So, since
$L_{11}\in GL(N[T]^{\perp},R[T])$, necessarily

$\mathscr{S}_{T,(P,Q)}(L)\in GL(N[T],R[T]^{\perp})$

and therefore $\mathcal{D}_{T,(P,Q)}(L)\neq 0$. To prove the converse, assume
that $\mathcal{D}_{T,(P,Q)}(L)=0$. Then,

$\mathscr{S}_{T,(P,Q)}(L)\notin GL(N[T],R[T]^{\perp})$

and hence,

$L_{11}\oplus\mathscr{S}_{T,(P,Q)}(L)\notin GL(U,V).$

Consequently, by (3.5), $L\notin GL(U,V)$. This concludes the proof. ∎

Thanks to this (local) notion of determinant, the generalized algebraic
multiplicity can be also expressed in the vein of (2.4) even in an infinite-
dimensional setting. Precisely, the next result holds.

######  Theorem 3.3.

Assume
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$,
i.e. $\mathfrak{L}\in\mathcal{C}^{r}(\Omega,\Phi_{0}(U,V)$ with
$\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathfrak{L})$ for some
$1\leq\kappa\leq r$ and $r\in\mathbb{N}$. Then,

(3.6)
$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\operatorname{ord}_{\lambda_{0}}\det\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))$

for every pair $(P,Q)$ of $\mathfrak{L}(\lambda_{0})$-projections.

###### Proof.

First, we will show that, for every pair $(P,Q)$ of
$\mathfrak{L}(\lambda_{0})$-projections,

(3.7)
$\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L}).$

Indeed, let $(P,Q)$ be a pair of $\mathfrak{L}(\lambda_{0})$-projections.
Then,

$\mathfrak{L}(\lambda)=\left(\begin{array}[]{cc}L_{11}(\lambda)&L_{12}(\lambda)\\\\[4.30554pt]
L_{21}(\lambda)&L_{22}(\lambda)\end{array}\right),$

where

$\begin{array}[]{ll}L_{11}(\lambda):=Q\mathfrak{L}(\lambda)(I_{U}-P),&L_{12}(\lambda):=Q\mathfrak{L}(\lambda)P,\\\\[4.30554pt]
L_{21}(\lambda):=(I_{V}-Q)\mathfrak{L}(\lambda)(I_{U}-P),&L_{22}(\lambda):=(I_{V}-Q)\mathfrak{L}(\lambda)P.\end{array}$

Since

$L_{11}(\lambda_{0})\in
GL(N[\mathfrak{L}(\lambda_{0})]^{\perp},R[\mathfrak{L}(\lambda_{0})]),$

we can take $\Omega_{\lambda_{0}}$ sufficiently small so that

$L_{11}(\lambda)\in
GL(N[\mathfrak{L}(\lambda_{0})]^{\perp},R[\mathfrak{L}(\lambda_{0})])\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}}.$

Thus, by (3.5), it becomes apparent that

(3.8)
$L_{11}(\lambda)\oplus\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\mathfrak{L}_{1}^{-1}(\lambda)\mathfrak{L}(\lambda)\mathfrak{L}^{-1}_{2}(\lambda)\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}},$

where

(3.9)
$\mathfrak{L}_{1}(\lambda):=\left(\\!\\!\begin{array}[]{cc}I_{R[\mathfrak{L}(\lambda_{0})]}&0\\\\[4.30554pt]
L_{21}(\lambda)L^{-1}_{11}(\lambda)&I_{R[\mathfrak{L}(\lambda_{0})]^{\perp}}\end{array}\\!\\!\right),\;\;\mathfrak{L}_{2}(\lambda):=\left(\\!\\!\begin{array}[]{cc}I_{N[\mathfrak{L}(\lambda_{0})]^{\perp}}&L^{-1}_{11}(\lambda)L_{12}(\lambda)\\\\[4.30554pt]
0&I_{N[\mathfrak{L}(\lambda_{0})]}\end{array}\\!\\!\right).$

Therefore, inverting (3.8) yields

(3.10)
${L}_{11}^{-1}(\lambda)\oplus[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]^{-1}=\mathfrak{L}_{2}(\lambda)\mathfrak{L}^{-1}(\lambda)\mathfrak{L}_{1}(\lambda)$

for all $\lambda\in\Omega_{\lambda_{0}}\setminus\\{\lambda_{0}\\}$. Since

$\mathfrak{L}(\Omega_{\lambda_{0}}\setminus\\{\lambda_{0}\\})\subset GL(U,V),$

by Theorem 3.2, we have that, for every
$\lambda\in\Omega_{\lambda_{0}}\setminus\\{\lambda_{0}\\}$,

$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\in
GL(N[\mathfrak{L}(\lambda_{0})],R[\mathfrak{L}(\lambda_{0})]^{\perp}).$

On the one hand, since
$\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathfrak{L})$, it is apparent that

(3.11)
$\|\mathfrak{L}_{2}(\lambda)\mathfrak{L}^{-1}(\lambda)\mathfrak{L}_{1}(\lambda)\|\leq
C\|\mathfrak{L}^{-1}(\lambda)\|\leq\frac{C}{|\lambda-\lambda_{0}|^{\kappa}}.$

Moreover,

(3.12)
$\|[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]^{-1}\|\leq\|L_{11}^{-1}(\lambda)\oplus[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]^{-1}\|.$

Consequently, taking norms in (3.10), we may conclude from (3.12) and (3.11)
that

$\|[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]^{-1}\|\leq\frac{C}{|\lambda-\lambda_{0}|^{k}}\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}}\setminus\\{\lambda_{0}\\},$

which ends the proof of (3.7).

According to (3.8), we already know that

(3.13)
$\mathfrak{L}(\lambda)=\mathfrak{L}_{1}(\lambda)\left[L_{11}(\lambda)\oplus\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\right]\mathfrak{L}_{2}(\lambda),$

with $\mathfrak{L}_{1}(\lambda),\mathfrak{L}_{2}(\lambda)\in GL(U,V)$ for all
$\lambda\in\Omega_{\lambda_{0}}$. Thus, by the properties of $\chi$ listed at
the end of Section 3, it is apparent that

$\chi[\mathfrak{L}_{1},\lambda_{0}]=\chi[\mathfrak{L}_{2},\lambda_{0}]=0.$

Moreover, since $L_{11}(\lambda)\in GL(U,V)$ for all
$\lambda\in\Omega_{\lambda_{0}}$,

$\displaystyle\chi[L_{11}(\lambda)\oplus(\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L}),\lambda_{0}]$
$\displaystyle=\chi[L_{11}(\lambda),\lambda_{0}]+\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L},\lambda_{0}]$
$\displaystyle=\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L},\lambda_{0}].$

Observe that
$\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),\mathcal{P}}\circ\mathfrak{L},\lambda_{0}]$
is well defined by (3.7). Therefore, applying the product formula to (3.13)
yields

$\chi[\mathfrak{L},\lambda_{0}]=\chi[\mathfrak{L}_{1},\lambda_{0}]+\chi[L_{11}\oplus(\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L}),\lambda_{0}]+\chi[\mathfrak{L}_{2},\lambda_{0}]=\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L},\lambda_{0}].$

Finally, since
$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L}$ is finite-
dimensional and $L_{11}(\lambda)$ is an isomorphism, it follows from (2.4) and
Definition 3.1 that

$\displaystyle\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L},\lambda_{0}]$
$\displaystyle=\operatorname{ord}_{\lambda_{0}}\mathcal{D}\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))$
$\displaystyle=\operatorname{ord}_{\lambda_{0}}\mathcal{D}\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\cdot\mathcal{D}L_{11}(\lambda)$
$\displaystyle=\operatorname{ord}_{\lambda_{0}}\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda)).$

Lastly, taking into account that

$\chi[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\det\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda)),$

the proof is completed. ∎

To conclude this section, we are going to complete the proof of Theorem 1.1.
Note that Theorem 3.3 establishes the validity of the first identity of (1.8).
Assume $\mathfrak{L}\in\mathcal{C}(\Omega,\Phi_{0}(U,V))$, let
$\lambda_{0}\in\Omega\cap\Sigma(\mathfrak{L})$ be an isolated eigenvalue, and
consider a pair $(P,Q)$ of $\mathfrak{L}(\lambda_{0})$-projections. According
to (3.8), we already know that

$\mathfrak{L}(\lambda)=\mathfrak{L}_{1}(\lambda)\left(\begin{array}[]{cc}L_{11}(\lambda)&0\\\\[4.30554pt]
0&\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\end{array}\right)\mathfrak{L}_{2}(\lambda),$

where

$\mathfrak{L}_{1}(\lambda):=\left(\begin{array}[]{cc}I_{R[\mathfrak{L}(\lambda_{0})]}&0\\\\[4.30554pt]
L_{21}(\lambda)L^{-1}_{11}(\lambda)&I_{R[\mathfrak{L}(\lambda_{0})]^{\perp}}\end{array}\right),\;\;\mathfrak{L}_{2}(\lambda):=\left(\begin{array}[]{cc}I_{N[\mathfrak{L}(\lambda_{0})]^{\perp}}&L^{-1}_{11}(\lambda)L_{12}(\lambda)\\\\[4.30554pt]
0&I_{N[\mathfrak{L}(\lambda_{0})]}\end{array}\right).$

Thus, in a perforated neighborhood of $\lambda_{0}$ the inverse
$\mathfrak{L}^{-1}(\lambda)$ is given through

$\left(\\!\\!\\!\begin{array}[]{cc}I_{N[\mathfrak{L}(\lambda_{0})]^{\perp}}&-L_{11}^{-1}(\lambda)L_{12}(\lambda)\\\\[4.30554pt]
0&I_{N[\mathfrak{L}(\lambda_{0})]}\end{array}\\!\\!\\!\right)\\!\\!\\!\left(\\!\\!\\!\begin{array}[]{cc}L_{11}^{-1}(\lambda)&0\\\\[4.30554pt]
0&\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\end{array}\\!\\!\\!\right)\\!\\!\\!\left(\\!\\!\\!\begin{array}[]{cc}I_{R[\mathfrak{L}(\lambda_{0})]}&0\\\\[4.30554pt]
-L_{21}(\lambda)L^{-1}_{11}(\lambda)&I_{R[\mathfrak{L}(\lambda_{0})]^{\perp}}\end{array}\\!\\!\\!\right).$

Multiplying these operator matrices, it becomes apparent that

$\mathfrak{L}^{-1}=\left(\begin{array}[]{cc}L_{11}^{-1}+L_{11}^{-1}L_{12}\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L})L_{21}L_{11}^{-1}&-L_{11}^{-1}L_{12}\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L})\\\
-\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L})L_{21}L_{11}^{-1}&\mathscr{S}^{-1}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L})\end{array}\right).$

Therefore, since

$\mathfrak{L}^{-1}(\lambda)=\left(\begin{array}[]{cc}Q\mathfrak{L}^{-1}(\lambda)(I_{U}-P)&Q\mathfrak{L}^{-1}(\lambda)P\\\\[4.30554pt]
(I_{V}-Q)\mathfrak{L}^{-1}(\lambda)(I_{U}-P)&(I_{V}-Q)\mathfrak{L}^{-1}(\lambda)P\end{array}\right),$

identifying the last entries of these two operator matrices, (1.7) holds.
Therefore, by Theorem 3.3, (1.8) follows readily if
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$.
This ends the proof of Theorem 1.1.

##  4\. Proof of Theorem 1.2

First, we will recover $\chi$ from the perspective of Nonlinear Analysis, by
adopting the point of view of Bifurcation Theory. Actually, this was the
methodology of Esquinas and López-Gómez [7] to construct $\chi$. In
applications one is naturally lead to deal with nonlinear equations of the
form

(4.1)
$\mathscr{F}(\lambda,u)\equiv\mathfrak{L}(\lambda)u+\mathfrak{N}(\lambda,u)=0,$

where $\mathfrak{L}\in\mathcal{C}^{1}(\Omega,\Phi_{0}(U,V))$ for some
subdomain $\Omega$ of $\mathbb{K}$, and
$\mathfrak{N}\in\mathcal{C}^{1}(\Omega\times U,V)$ satisfies

(4.2) $\mathfrak{N}(\lambda,0)=0\quad\hbox{and}\quad
D_{u}\mathfrak{N}(\lambda,0)=0$

for all $\lambda$ in a neighborhood of a given $\lambda_{0}\in\Omega$. Then,
$\mathscr{F}(\lambda,0)=0$ for $\lambda$ sufficiently close to $\lambda_{0}$
and, for any given pair $(P,Q)$ of $\mathfrak{L}(\lambda_{0})$-projections,
the equation (4.1) can be equivalently written as

(4.3)
$\left\\{\begin{array}[]{l}Q\mathfrak{L}(\lambda)(x+y)+Q\mathfrak{N}(\lambda,x+y)=0,\\\\[4.30554pt]
(I_{V}-Q)\mathfrak{L}(\lambda)(x+y)+(I_{V}-Q)\mathfrak{N}(\lambda,x+y)=0,\end{array}\right.$

where we are setting

$x:=Pu\in N[\mathfrak{L}(\lambda_{0})],\quad y:=(I_{U}-P)u\in R[I_{U}-P]\equiv
N[\mathfrak{L}(\lambda_{0})]^{\perp}.$

Subsequently, we consider the map $\mathscr{G}$ defined by

$\mathscr{G}(\lambda,x,y):=Q\mathfrak{L}(\lambda)(x+y)+Q\mathfrak{N}(\lambda,x+y)$

for every $(\lambda,x,y)$ in a neighborhood, $\mathcal{O}$, of
$(\lambda_{0},0,0)$ in

$\Omega\times N[\mathfrak{L}(\lambda_{0})]\times
N[\mathfrak{L}(\lambda_{0})]^{\perp}\thickapprox\Omega\times U.$

The map $\mathscr{G}$ is of class $\mathcal{C}^{1}(\mathcal{O})$ and it
satisfies

$D_{y}\mathscr{G}(\lambda,0,0)=Q\mathfrak{L}(\lambda)(I_{U}-P)\equiv
L_{11}(\lambda)$

for $\lambda$ in a certain neighborhood, $\tilde{\Omega}\subset\Omega$, of
$\lambda_{0}$, where we can assume that

$L_{11}(\lambda)\in
GL(N[\mathfrak{L}(\lambda_{0})]^{\perp},R[\mathfrak{L}(\lambda_{0})]).$

Moreover, $\mathscr{G}(\lambda,0,0)=0$ for all $\lambda\in\tilde{\Omega}$.
Thus, since

$D_{y}\mathscr{G}(\lambda,0,0)=L_{11}(\lambda)\in
GL(N[\mathfrak{L}(\lambda_{0})]^{\perp},R[\mathfrak{L}(\lambda_{0})]),$

by the Implicit Function Theorem, there exist a neighborhood, $\mathcal{U}$,
of $(\lambda_{0},0)$ in $\mathbb{K}\times N[\mathfrak{L}(\lambda_{0})]$ and a
map $\mathscr{Y}\in\mathcal{C}^{1}(\mathcal{U},R[\mathfrak{L}(\lambda_{0})])$
such that

$\mathscr{Y}(\lambda,0)=0\quad\hbox{and}\quad\mathscr{G}(\lambda,x,\mathscr{Y}(\lambda,x))=0$

for all $(\lambda,x)\in\mathcal{U}$, and these are the unique zeroes of
$\mathscr{G}$ in $\mathcal{U}$. So, $\mathscr{Y}(\lambda,0)=0$ for $\lambda$
sufficiently close to $\lambda_{0}$, because $\mathscr{G}(\lambda,0,0)=0$.
Furthermore, by the definition of $\mathscr{G}$, we have that, for every
$(\lambda,x)\in\mathcal{U}$,

(4.4)
$Q\mathfrak{L}(\lambda)(x+\mathscr{Y}(\lambda,x))+Q\mathfrak{N}(\lambda,x+\mathscr{Y}(\lambda,x))=0.$

Thus, thanks to (4.2), differentiating (4.4) with respect to $x$ yields to

$Q\mathfrak{L}(\lambda)(x+D_{x}\mathscr{Y}(\lambda,0)x)=0\quad\hbox{for
all}\;\;x\in N[\mathfrak{L}(\lambda_{0})].$

Equivalently,

$Q\mathfrak{L}(\lambda)(I_{U}-P)D_{x}\mathscr{Y}(\lambda,0)x=-Q\mathfrak{L}(\lambda)Px\quad\hbox{for
all}\;\;x\in N[\mathfrak{L}(\lambda_{0})].$

Consequently, for sufficiently small $\tilde{\Omega}$, we find that, for every
$\lambda\in\tilde{\Omega}$,

(4.5)
$D_{x}\mathscr{Y}(\lambda,0)=-\left[Q\mathfrak{L}(\lambda)(I_{U}-P)\right]^{-1}Q\mathfrak{L}(\lambda)P=-L_{11}^{-1}(\lambda)L_{12}(\lambda).$

Consequently, substituting $y$ by $\mathscr{Y}(\lambda,x)$ into the second
equation of (4.3), it turns out that solving (4.1) locally at
$(\lambda,u)=(\lambda_{0},0)$ is equivalent to solve the equation

(4.6)
$(I_{V}-Q)\mathfrak{L}(\lambda)(x+\mathscr{Y}(\lambda,x))+(I_{V}-Q)\mathfrak{N}(\lambda,x+\mathscr{Y}(\lambda,x))=0$

in a neighborhood of $(\lambda_{0},0)$ in $\mathbb{K}\times
N[\mathfrak{L}(\lambda_{0})]$. Equation (4.6) is often referred to as the
_bifurcation equation_ of (4.1) at $(\lambda_{0},0)$. According to [20, Le.
3.1.1], $(\lambda,u)=(\lambda_{0},0)$ is a bifurcation point of (4.1) from
$(\lambda,u)=(\lambda,0)$ if and only if $(\lambda,x)=(\lambda_{0},0)$ is a
bifurcation point of (4.5) from $(\lambda,x)=(\lambda,0)$. As the
linearization of the left hand side of (4.6) at $(\lambda,x)=(\lambda,0)$ is
given by the linear operator

$\mathscr{B}(\lambda)x:=(I_{V}-Q)\mathfrak{L}(\lambda)\left(x+D_{x}\mathscr{Y}(\lambda,0)x\right)\quad\hbox{for
all}\;\;x\in N[\mathfrak{L}(\lambda)],$

and, due to (4.5), $\mathscr{B}(\lambda)$ can be expressed as

$\mathscr{B}(\lambda):=(I_{V}-Q)\mathfrak{L}(\lambda)\left[P-(I_{U}-P)L_{11}^{-1}(\lambda)L_{12}(\lambda)\right],$

by the definitions of the $L_{ij}(\lambda)$’s, it becomes apparent that

(4.7)
$\mathscr{B}(\lambda):=L_{22}(\lambda)-L_{21}(\lambda)L_{11}^{-1}(\lambda)L_{12}(\lambda)=\mathscr{S}_{\mathfrak{L}(\lambda),(P,Q)}(\mathfrak{L}(\lambda)).$

Therefore, the first part of Theorem 1.2 and, in particular, (1.9) is proven.
Naturally, (1.10) follows readily by combining (1.9) with Theorem 1.1. The
equivalence between (b), (c), (d), and (e) is a direct consequence from (1.8)
and (1.10), and the equivalence of (a) and (b) is [20, Th. 4.3.4].

Finally, suppose that $\chi[\mathfrak{L},\lambda_{0}]$ is an odd integer.
Then, due to (1.10), $\det\mathscr{B}(\lambda)$ changes sign as $\lambda$
crosses $\lambda_{0}$. Hence, also the Brouwer degree of $(\lambda,0)$ changes
as $\lambda$ crosses $\lambda_{0}$. Thus, thanks to [20, Th. 6.2.1], the
bifurcation equation (4.6) possesses a (local) continuum of non-trivial
solutions emanating from $(\lambda,x)=(\lambda,0)$ at $\lambda_{0}$.
Naturally, by the local structure of the solutions of (4.1) in a neighborhood
of $(\lambda,u)=(\lambda_{0},0)$, it becomes apparent that (4.1) also has a
(local) continuum of non-trivial solutions emanating from
$(\lambda,u)=(\lambda,0)$ at $\lambda=\lambda_{0}$. Consequently, Zorn’s lemma
shows the existence of the component $\mathscr{C}$. This ends the proof of
Theorem 1.2.

##  5\. The classical multiplicity from a geometrical perspective

The main goal of this section is establishing a, rather hidden, bi-association
between the classical algebraic multiplicity and the concept of intersection
index of varieties in Algebraic Geometry. Naturally, this might open new
challenging perspectives in both fields, besides incrementing their number of
potential applications. The next, and final, section will extend the findings
of this one to cover the generalized algebraic multiplicity $\chi$. As for
algebraic purposes one needs to work in algebraically closed fields,
throughout the rest of this paper we will assume that $\mathbb{K}=\mathbb{C}$.

Subsequently, we identify
$\mathcal{L}(\mathbb{C}^{N})\simeq\mathbb{C}^{N^{2}}$ and regard the
determinant map,

$\mathcal{D}:\mathcal{L}(\mathbb{C}^{N})\to\mathbb{C},\qquad\mathcal{D}(T):=\det
T\;\;\hbox{for every}\;\;T\in\mathcal{L}(\mathbb{C}^{N}),$

as an homogeneous polynomial map of degree $N$. Using this identification,
every $T\in\mathcal{L}(\mathbb{C}^{N})$ can be expressed as en element of
$\mathbb{C}^{N^{2}}$ through the identification

$T=\left(\begin{array}[]{cccc}x_{1}&x_{2}&\cdots&x_{N}\\\
x_{N+1}&x_{N+2}&\cdots&x_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{N^{2}-N+1}&x_{N^{2}-N+2}&\cdots&x_{N^{2}}\end{array}\right)=\left(x_{1},x_{2},\ldots,x_{N^{2}-1},x_{N^{2}}\right).$

Then, for every $1\leq k\leq N^{2}$, we define the $k$-th differential
functional of the determinant map at a given operator
$L\in\mathcal{L}(\mathbb{C}^{N})$ as the map
$\mathcal{D}^{k}(L):\mathcal{L}(\mathbb{C}^{N})\to\mathbb{C}$ defined by

$\mathcal{D}^{k}(L)[T]:=\sum_{1\leq j_{1},j_{2},\cdots,j_{k}\leq
N^{2}}\left(\frac{\partial^{k}\mathcal{D}}{\partial x_{j_{1}}\partial
x_{j_{2}}\cdots\partial x_{j_{k}}}\right)(L)\cdot x_{j_{1}}x_{j_{2}}\cdots
x_{j_{k}}$

for all $T\in\mathcal{L}(\mathbb{C}^{N})$. Note that, for any given
$L\in\mathcal{L}(\mathbb{C}^{N})$, the Taylor expansion of the determinant
map, $\mathcal{D}$, centered at $L$, is given through

$\mathcal{D}(T)=\mathcal{D}(L)+\sum_{k=1}^{N^{2}}\frac{1}{k!}\mathcal{D}^{k}(L)[T-L]\qquad\hbox{for
all}\;\;T\in\mathcal{L}(\mathbb{C}^{N}).$

In particular, $\mathcal{D}^{k}(L)$ can be viewed as an homogeneous polynomial
of degree $k$. The associated algebraic variety

$V(\mathcal{D}^{k}(L))=(\mathcal{D}^{k}(L))^{-1}(0)$

will be subsequently refereed to as the $k$_-th tangent variety_ of the
determinant map at $L\in\mathcal{L}(\mathbb{C}^{N})$, and will be denoted by
$\mathscr{T}_{L}^{k}\mathcal{D}$. As $\mathscr{T}^{1}_{L}\mathcal{D}$ is the
tangent space to $D^{-1}(0)$ at $L$, the variety
$\mathscr{T}^{k}_{L}\mathcal{D}$ can be viewed as a generalization of the
classical tangent space at a given point. The following technical result can
be easily obtained from the chain rule.

######  Lemma 5.1.

Let $T\in\mathcal{L}(\mathbb{C}^{N})$ and $\mathfrak{L}(\lambda)=\lambda
I_{N}-T$ for all $\lambda\in\mathbb{C}$. Then, for every $r\in\mathbb{N}$, the
following identity holds

(5.1)
$\frac{d^{r}}{d\lambda^{r}}(\mathcal{D}\mathfrak{L}(\lambda))\equiv(\mathcal{D}\mathfrak{L})^{(r)}(\lambda)=\mathcal{D}^{r}(\mathfrak{L}(\lambda))[I_{N}].$

###### Proof.

For notational convenience, we will identify $x_{i,j}$ with $x_{(i-1)N+j}$ for
any $N\times N$ matrix $(x_{i,j})_{1\leq i,j\leq N}$. By definition,

$\mathcal{D}^{r}(\mathfrak{L}(\lambda))[I_{N}]=\sum_{1\leq
i_{1},i_{2},\cdots,i_{r}\leq N}\left(\frac{\partial^{r}\mathcal{D}}{\partial
x_{i_{1},i_{1}}\partial x_{i_{2},i_{2}}\cdots\partial
x_{i_{r},i_{r}}}\right)(\mathfrak{L}(\lambda)).$

Thus, (5.1) can be equivalently written in the form

(5.2) $(\mathcal{D}\mathfrak{L})^{(r)}(\lambda)=\sum_{1\leq
i_{1},i_{2},\cdots,i_{r}\leq N}\left(\frac{\partial^{r}\mathcal{D}}{\partial
x_{i_{1},i_{1}}\partial x_{i_{2},i_{2}}\cdots\partial
x_{i_{r},i_{r}}}\right)(\mathfrak{L}(\lambda)).$

The proof of (5.2) will proceed by induction on $r$. Suppose $r=1$. Then,
since $\mathfrak{L}^{\prime}(\lambda)=I_{N}$, we have that

$(\mathcal{D}\mathfrak{L})^{{}^{\prime}}(\lambda)=\nabla\mathcal{D}(\mathfrak{L}(\lambda))\circ\mathfrak{L}^{\prime}(\lambda)=\sum_{i=1}^{N^{2}}\frac{\partial\mathcal{D}}{\partial
x_{i}}(\mathfrak{L}(\lambda))\cdot[\mathfrak{L}^{\prime}(\lambda)]_{i}=\sum_{i_{1}=1}^{N}\frac{\partial\mathcal{D}}{\partial
x_{i_{1},i_{1}}}(\mathfrak{L}(\lambda)),$

where $[\mathfrak{L}^{\prime}(\lambda)]_{i}$ is the $i$-th term of the matrix
$\mathfrak{L}^{\prime}(\lambda)$. Thus, (5.2) holds for $r=1$. Now, suppose
that (5.2) is satisfied for some $r=n-1$ with $n\geq 2$, i.e.,

(5.3) $(\mathcal{D}\mathfrak{L})^{(n-1)}(\lambda)=\sum_{1\leq
i_{1},i_{2},\cdots,i_{n-1}\leq
N}\left(\frac{\partial^{n-1}\mathcal{D}}{\partial x_{i_{1},i_{1}}\partial
x_{i_{2},i_{2}}\cdots\partial
x_{i_{n-1},i_{n-1}}}\right)(\mathfrak{L}(\lambda)).$

Then, differentiating with respect to $\lambda$ yields

$\displaystyle(\mathcal{D}\mathfrak{L})^{(n)}(\lambda)$
$\displaystyle=\frac{d}{d\lambda}\sum_{1\leq i_{1},i_{2},\cdots,i_{n-1}\leq
N}\left(\frac{\partial^{n-1}\mathcal{D}}{\partial x_{i_{1},i_{1}}\partial
x_{i_{2},i_{2}}\cdots\partial
x_{i_{n-1},i_{n-1}}}\right)(\mathfrak{L}(\lambda))$ $\displaystyle=\sum_{1\leq
i_{1},i_{2},\cdots,i_{n-1}\leq
N}\frac{d}{d\lambda}\left[\left(\frac{\partial^{n-1}\mathcal{D}}{\partial
x_{i_{1},i_{1}}\partial x_{i_{2},i_{2}}\cdots\partial
x_{i_{n-1},i_{n-1}}}\right)(\mathfrak{L}(\lambda))\right]$
$\displaystyle=\sum_{1\leq i_{1},i_{2},\cdots,i_{n-1}\leq
N}\sum_{j=1}^{N^{2}}\left(\frac{\partial^{n}\mathcal{D}}{\partial
x_{i_{1},i_{1}}\partial x_{i_{2},i_{2}}\cdots\partial
x_{i_{n-1},i_{n-1}}\partial
x_{j}}\right)(\mathfrak{L}(\lambda))\cdot[\mathfrak{L}^{\prime}(\lambda)]_{j}$
$\displaystyle=\sum_{1\leq i_{1},i_{2},\cdots,i_{n}\leq
N}\left(\frac{\partial^{n}\mathcal{D}}{\partial x_{i_{1},i_{1}}\partial
x_{i_{2},i_{2}}\cdots\partial x_{i_{n},i_{n}}}\right)(\mathfrak{L}(\lambda)).$

This concludes the proof. ∎

The following result, illustrated by Figure 1, gives a geometrical meaning to
the classical algebraic multiplicity of an eigenvalue of a linear operator. As
illustrated by Figure 1, the algebraic multiplicity provides us with the
minimal integer $\mathfrak{m}$ for which the straight line $\lambda I_{N}$ is
not contained within the $\mathfrak{m}$-th tangent variety of the determinant
map at $\mathfrak{L}(\lambda_{0})$, which has been denoted by
$\mathscr{T}^{\mathfrak{m}}_{\mathfrak{L}(\lambda_{0})}\mathcal{D}$. Actually,
Figure 1 illustrates an admissible case when $\lambda I_{N}$ lies entirely
within the tangent variety
$\mathscr{T}^{1}_{\mathfrak{L}(\lambda_{0})}\mathcal{D}$ and hence,

$\mathfrak{m}=\chi[\lambda I_{N}-T,\lambda_{0}]\geq 2.$

######  Theorem 5.2.

Let $T\in\mathcal{L}(\mathbb{C}^{N})$, pick $\lambda_{0}\in\sigma(T)$, and
consider the straight line

$\mathfrak{L}(\lambda)=\lambda I_{N}-T,\quad\lambda\in\mathbb{C}.$

Then, $\mathfrak{m}:=\chi[\mathfrak{L},\lambda_{0}]$ is the minimal integer
$\mathfrak{m}\geq 1$ for which the line $\lambda I_{N}$ is contained in
$\mathscr{T}^{i}_{\mathfrak{L}(\lambda_{0})}\mathcal{D}$ for every $1\leq
i<\mathfrak{m}$, but not for $i=\mathfrak{m}$.

###### Proof.

By (5.3), $\mathfrak{m}=\chi[\mathfrak{L},\lambda_{0}]$ if and only if

$(\mathcal{D}\mathfrak{L})^{(i)}(\lambda_{0})=0\;\;\text{ for each
}\;\;i\in\\{1,...,\mathfrak{m}-1\\}\;\;\hbox{and}\;\;(\mathcal{D}\mathfrak{L})^{(\mathfrak{m})}(\lambda_{0})\neq
0.$

Therefore, owing to (5.1), it becomes apparent that

$\mathcal{D}^{i}(\mathfrak{L}(\lambda_{0}))[I_{N}]=0\;\;\text{ for each
}\;\;i\in\\{1,...,\mathfrak{m}-1\\}\;\;\hbox{and}\;\;\mathcal{D}^{\mathfrak{m}}(\mathfrak{L}(\lambda_{0}))[I_{N}]\neq
0.$

We conclude the proof by noticing that the line $\lambda I_{N}$ is contained
in the variety

$\mathscr{T}^{i}_{\mathfrak{L}(\lambda_{0})}\mathcal{D}=\left(\mathcal{D}^{i}(\mathfrak{L}(\lambda_{0}))\right)^{-1}(0)$

if and only if

$\mathcal{D}^{i}(\mathfrak{L}(\lambda_{0}))[\lambda I_{N}]=0\text{ for all
}\lambda\in\mathbb{C}.$

As the map $\mathcal{D}^{i}(\mathfrak{L}(\lambda_{0}))$ is linear, this occurs
if and only if

$\mathcal{D}^{i}(\mathfrak{L}(\lambda_{0}))[I_{N}]=0.$

The proof is complete. ∎

$\mathcal{D}^{-1}(0)$$\mathfrak{L}(\lambda_{0})$$\mathcal{T}^{1}_{\mathfrak{L}(\lambda_{0})}\mathcal{D}$$\mathfrak{L}$$\mathcal{L}\left(\mathbb{C}^{N}\right)$
Figure 1. Geometrical Meaning of Classical Algebraic Multiplicity

Theorem 5.2 leads us to think of the existence of a sharp connection between
Spectral Theory and Algebraic Geometry, which had been already stated in
Theorem 1.3. The rest of this section is devoted to deliver a proof of this
theorem.

Let $T\in\mathcal{L}(\mathbb{C}^{N})$, pick $\lambda_{0}\in\sigma(T)$, and
consider the straight line

$\mathfrak{L}(\lambda)=\lambda I_{N}-T,\quad\lambda\in\mathbb{C}.$

As usual in this section, we express the operator
$T\in\mathcal{L}(\mathbb{C}^{N})$ in matrix form

$T\sim\left(\begin{array}[]{cccc}t_{1}&t_{2}&\cdots&t_{N}\\\
t_{N+1}&t_{N+2}&\cdots&t_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\
t_{N^{2}-N+1}&t_{N^{2}-N+2}&\cdots&t_{N^{2}}\end{array}\right),$

i.e., setting $T=\left(t_{i,j}\right)_{1\leq i,j\leq N}$, we are identifying
$t_{i,j}$ with $t_{(i-1)N+j}$.

Naturally, the parametric equations of the line $\mathfrak{L}(\mathbb{C})$ in
$\mathbb{C}^{N^{2}}$ are the following ones:

$\left\\{\begin{array}[]{ll}x_{i}=\lambda-t_{i}&\text{ for
}i\in\\{(m-1)N+m:m\in\\{1,2,...,N\\}\\},\\\ x_{i}=-t_{i}&\text{ for
}i\in\\{1,2,...,N^{2}\\}\backslash\\{(m-1)N+m:m\in\\{1,2,...,N\\}\\}.\end{array}\right.$

Thus, by a simple computation involving the following identity

(5.4) $\lambda=x_{1}+t_{1},$

it is easily seen that $\mathfrak{L}(\mathbb{C})$ is the algebraic variety
generated by the polynomials

$\left\\{\begin{array}[]{ll}\mathfrak{P}_{i-1}\equiv
x_{i}-x_{1}+t_{i}-t_{1}&\text{ for }i\in\\{(m-1)N+m:m\in\\{2,...,N\\}\\},\\\
\mathfrak{P}_{i-1}\equiv x_{i}+t_{i}&\text{ for
}i\in\\{1,2,...,N^{2}\\}\backslash\\{(m-1)N+m:m\in\\{1,2,...,N\\}\\}.\end{array}\right.$

In other words,

$\mathfrak{L}(\mathbb{C})=V\left(\mathfrak{P}_{1},\ldots,\mathfrak{P}_{N^{2}-1}\right),$

i.e., the implicit equations of $\mathfrak{L}(\mathbb{C})$ are

$\left\\{\begin{array}[]{ll}x_{i}-x_{1}+t_{i}-t_{1}=0&\text{ for
}i\in\\{(m-1)N+m:m\in\\{2,...,N\\}\\},\\\ x_{i}+t_{i}=0&\text{ for
}i\in\\{1,2,...,N^{2}\\}\backslash\\{(m-1)N+m:m\in\\{1,2,...,N\\}\\}.\end{array}\right.$

According to Theorem 18.18 of Eisenbud [4], the determinantal variety
$\mathcal{D}^{-1}(0)$ is Cohen-Macaulay, as well as
$\mathfrak{L}(\mathbb{C})$, because it is smooth (see [4, Sect. 18.2]).
Therefore, by Eisenbud and Harris [5, p. 48],

$i(\mathcal{D}^{-1}(0),\mathfrak{L}(\mathbb{C});\mathfrak{L}(\lambda_{0}))=\ell_{\mathcal{O}}(\mathcal{O}/(\mathfrak{D},\mathfrak{P}))$

(see (1.12)), where
$\mathcal{O}\equiv\mathcal{O}_{\mathfrak{L}(\lambda_{0}),\mathbb{C}^{N^{2}}}$
stands for the local ring of $\mathbb{C}^{N^{2}}$ at
$\mathfrak{L}(\lambda_{0})$, $\mathfrak{D}$ is the ideal generated by the
determinant map,

$\mathfrak{D}:=(\mathcal{D}(x_{1},x_{2},\cdots,x_{N^{2}})),$

$\mathfrak{P}$ stands for the ideal generated by the polynomials
$\mathfrak{P}_{j}$, $1\leq j\leq N^{2}-1$, and $\ell_{\mathcal{O}}$ is the
length of the module $\mathcal{O}/(\mathfrak{D},\mathfrak{P})$ over the ring
$\mathcal{O}$. By (5.4),

$\displaystyle(\mathfrak{D},\mathfrak{P})$
$\displaystyle=(\mathcal{D}(x_{1},x_{2},\cdots,x_{N^{2}}),\mathfrak{P}_{1},\mathfrak{P}_{2},\cdots,\mathfrak{P}_{N^{2}-1})$
$\displaystyle=(\mathcal{D}[(x_{1}+t_{1})I_{N}-T],\mathfrak{P}_{1},\mathfrak{P}_{2},\cdots,\mathfrak{P}_{N^{2}-1})$
$\displaystyle=(\mathcal{D}\mathfrak{L}(x_{1}+t_{1}),\mathfrak{P}_{1},\mathfrak{P}_{2},\cdots,\mathfrak{P}_{N^{2}-1}).$

On the other hand, by the definition of the classical algebraic geometry,
$\mathfrak{m}=\chi[\lambda I_{N}-T,\lambda_{0}]$, the following identity holds

$\mathcal{D}\mathfrak{L}(x_{1}+t_{1})=(x_{1}+t_{1}-\lambda_{0})^{\mathfrak{m}}f(x_{1}+t_{1})$

for some polynomial $f$ such that $f(\lambda_{0})\neq 0$. Thus, since
$f(x_{1}+t_{1})$ is a unit in the local ring
$\mathcal{O}_{\mathfrak{L}(\lambda_{0}),\mathbb{C}^{N^{2}}}$, we find that

$(\mathfrak{D},\mathfrak{P})=((x_{1}+t_{1}-\lambda_{0})^{\mathfrak{m}},\mathfrak{P}_{1},\mathfrak{P}_{2},\cdots,\mathfrak{P}_{N^{2}-1}).$

On the other hand, by a routine calculation,

$V((x_{1}+t_{1}-\lambda_{0})^{\mathfrak{m}},\mathfrak{P}_{1},\mathfrak{P}_{2},\cdots,\mathfrak{P}_{N^{2}-1})=\left\\{\mathfrak{L}(\lambda_{0})\right\\}.$

Therefore, by, e.g., Proposition 6 of Fulton [9], we get the isomorphism

$\frac{\mathcal{O}_{\mathfrak{L}(\lambda_{0}),\mathbb{C}^{N^{2}}}}{(\mathfrak{D},\mathfrak{P})}\simeq\frac{\mathbb{C}[x_{1},x_{2},...,x_{N^{2}}]}{(\mathfrak{D},\mathfrak{P})}.$

Moreover, since the residue field of $\mathcal{O}$ is $\mathbb{C}$ and
$\mathfrak{m}\cdot\mathcal{O}/(\mathfrak{D},\mathfrak{P})=0$, where
$\mathfrak{m}$ is the maximal ideal of $\mathcal{O}$, necessarily, by
Proposition 5.9 of Bump [2],

$\ell_{\mathcal{O}}(\mathcal{O}/(\mathfrak{D},\mathfrak{P}))=\dim_{\mathbb{C}}(\mathcal{O}/(\mathfrak{D},\mathfrak{P})).$

Since in $\mathbb{C}[x_{1},x_{2},...,x_{N^{2}}]/(\mathfrak{D},\mathfrak{P})$
the following $N^{2}$ identities hold true

$\left\\{\begin{array}[]{ll}(x_{1}+t_{1}-\lambda_{0})^{\mathfrak{m}}=0&\\\
x_{i}=x_{1}+t_{1}-t_{i}&\text{ for }i\in\\{(m-1)n+m:m\in\\{2,...,n\\}\\},\\\
x_{i}=-t_{i}&\text{ for
}i\in\\{1,2,...,n^{2}\\}\backslash\\{(m-1)n+m:m\in\\{1,2,...,n\\}\\},\end{array}\right.$

it is apparent, from the Newton binomial, that $x_{1}^{\mathfrak{m}}$ depends
linearly on $x_{1}^{j}$, $0\leq j\leq\mathfrak{m}-1$, as well as the $x_{i}$’s
for all $i\in\\{2,...,N^{2}\\}$. Therefore,

$\frac{\mathbb{C}[x_{1},x_{2},...,x_{N^{2}}]}{(\mathfrak{D},\mathfrak{P})}=\langle
1,x_{1},x^{2}_{1},\cdots,x^{\mathfrak{m}-1}_{1}\rangle_{\mathbb{C}}$

and consequently,

$\displaystyle
i(\mathcal{D}^{-1}(0),\mathfrak{L}(\mathbb{R});\mathfrak{L}(\lambda_{0}))$
$\displaystyle=\ell_{\mathcal{O}}(\mathcal{O}/(\mathfrak{D},\mathfrak{P}))=\dim_{\mathbb{C}}(\mathcal{O}/(\mathfrak{D},\mathfrak{P}))$
$\displaystyle=\dim_{\mathbb{C}}\left(\mathbb{C}[x_{1},x_{2},...,x_{N^{2}}]/(\mathfrak{D},\mathfrak{P})\right)=\mathfrak{m}.$

This ends the proof of Theorem 1.3.

##  6\. Proof of Theorem 1.4

Thanks to the existence of the local Smith form for general analytic curves,
$\mathfrak{L}$, at isolated singular values, $\lambda_{0}$, the algebraic
multiplicity of these curves, $\chi[\mathfrak{L},\lambda_{0}]$, can be also
expressed as the intersection index of two algebraic varieties, as already
stated by Theorem 1.4. This section consists of the proof of that theorem.

Let
$\mathfrak{L}\in\mathcal{A}_{\lambda_{0}}(\Omega_{\lambda_{0}},\Phi_{0}(X,Y))$
be, and fix a pair $(P,Q)$ of $\mathfrak{L}(\lambda_{0})$-projections. Then,
thanks to Theorem 3.3, we have that

$\displaystyle\chi[\mathfrak{L},\lambda_{0}]$
$\displaystyle=\operatorname{ord}_{\lambda_{0}}\mathcal{D}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\operatorname{ord}_{\lambda_{0}}\mathcal{D}\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))$
$\displaystyle=\operatorname{ord}_{\lambda_{0}}\mathcal{D}[L_{22}(\lambda)-L_{21}(\lambda)L_{11}^{-1}(\lambda)L_{12}(\lambda)].$

For the sake of notation, let us identify

$\mathcal{L}(N[\mathfrak{L}(\lambda_{0})],R[\mathfrak{L}(\lambda_{0})]^{\perp})\simeq\mathcal{L}(\mathbb{C}^{N})\quad\hbox{with}\quad
N:=\dim N[\mathfrak{L}(\lambda_{0})].$

According to (3.7), we already know that

$\lambda_{0}\in\operatorname{Alg}_{\kappa}(\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}\circ\mathfrak{L})$

for some $1\leq\kappa\leq r$, where $\mathcal{C}^{r}$ is the class of
regularity of $\mathfrak{L}(\lambda)$. Thus, by Theorem 7.8.1 of López-Gómez
and Mora-Corral [23],
$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),\mathcal{P}}\circ\mathfrak{L}$ admits
a local Smith form at $\lambda_{0}$, i.e., there exist a neighborhood,
$\Omega_{\lambda_{0}}$, of $\lambda_{0}$ such that
$\Omega_{\lambda_{0}}\subset\Omega$, two invertible curves
$\mathfrak{E},\mathfrak{F}\in\mathcal{C}(\Omega_{\lambda_{0}},GL(\mathbb{C}^{N}))$,
and $N$ positive integers

$\kappa_{1}\geq\cdots\geq\kappa_{N}\geq 1$

such that

(6.1)
$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))=\mathfrak{E}(\lambda)\cdot\mathfrak{P}(\lambda)\cdot\mathfrak{F}(\lambda)\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}},$

where $\mathfrak{P}(\lambda)$ is the matrix of order $n$ defined by

(6.2)
$\mathfrak{P}(\lambda):=\mathrm{diag\,}\left((\lambda-\lambda_{0})^{\kappa_{1}},\ldots,(\lambda-\lambda_{0})^{\kappa_{N}}\right),$

which is the _local Smith form_ of the _Schur operator_ of
$\mathfrak{L}(\lambda)$ at $\lambda_{0}$.

On the other hand, according to Lemma 10.1.1 of López-Gómez and Mora-Corral
[23], there exist an integer $M\geq N$, two invertible matrix polynomial
curves
$\mathfrak{P}_{1},\mathfrak{P}_{2}\in\mathcal{H}(\Omega_{\lambda_{0}},GL(\mathbb{C}^{M}))$,
and an operator $\mathscr{L}\in\mathcal{L}(\mathbb{C}^{M})$ such that

(6.3) $\mathfrak{P}(\lambda)\oplus
I_{M-N}=\mathfrak{P}_{1}(\lambda)\cdot(\lambda
I_{M}-\mathscr{L})\cdot\mathfrak{P}_{2}(\lambda)\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}}.$

Hence, bringing together (6.1) and (6.3), it becomes apparent that there are
two finite dimensional curves
$\mathfrak{E}_{1},\mathfrak{F}_{1}\in\mathcal{C}(\Omega_{\lambda_{0}},GL(\mathbb{C}^{M}))$
such that

(6.4)
$\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))\oplus
I_{M-N}=\mathfrak{C}_{1}(\lambda)\cdot(\lambda
I_{M}-\mathscr{L})\cdot\mathfrak{F}_{1}(\lambda)\quad\hbox{for
all}\;\;\lambda\in\Omega_{\lambda_{0}}.$

Consequently, taking determinants on both sides of (6.4) it is apparent that

$\displaystyle\mathcal{D}[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]$
$\displaystyle=\mathcal{D}(\mathfrak{E}_{1}(\lambda)\cdot(\lambda
I_{M}-\mathscr{L})\cdot\mathfrak{F}_{1}(\lambda))$
$\displaystyle=\mathcal{D}(\mathfrak{E}_{1}(\lambda))\cdot\mathcal{D}(\lambda
I_{M}-\mathscr{L})\cdot\mathcal{D}(\mathfrak{F}_{1}(\lambda)).$

Thus, by Theorem 1.1,

$\chi[\mathfrak{L},\lambda_{0}]=\operatorname{ord}_{\lambda_{0}}\mathcal{D}[\mathscr{S}_{\mathfrak{L}(\lambda_{0}),(P,Q)}(\mathfrak{L}(\lambda))]=\operatorname{ord}_{\lambda_{0}}\mathcal{D}(\lambda
I_{M}-\mathscr{L}).$

On the other hand, owing to Theorem 1.3, we already know that

$\mathcal{D}(\lambda I_{M}-\mathscr{L})=i(\mathcal{D}^{-1}(0),\lambda
I_{M}-\mathscr{L};\lambda_{0}I_{M}-\mathscr{L}).$

Therefore,

$\chi[\mathfrak{L},\lambda_{0}]=i(\mathcal{D}^{-1}(0),\lambda
I_{M}-\mathscr{L};\lambda_{0}I_{M}-\mathscr{L}),$

which ends the proof of Theorem 1.4.

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