New Analytical and Geometrical Aspects of the Algebraic Multiplicity

TL;DR

This paper explores new analytical and geometrical properties of the algebraic multiplicity, linking it to algebraic geometry concepts and providing a deeper geometric understanding of spectral theory.

Contribution

It establishes a novel connection between algebraic multiplicity and local intersection index, enriching the geometric interpretation of spectral properties.

Findings

Link between algebraic multiplicity and intersection index

Introduction of local determinant of Schur operator

Deep geometrical meaning for algebraic multiplicity

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, $\mathfrak{L}(\lambda)$.