New Analytical and Geometrical Aspects of the Algebraic Multiplicity

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