Mixed Segre zeta functions and their log-concavity

TL;DR

This paper introduces the mixed Segre zeta function for sequences of homogeneous ideals, proving its rationality, dependence on integral closure, and Lorentzian properties, thereby unifying and extending prior results in algebraic geometry.

Contribution

It generalizes and unifies previous work on Segre classes and zeta functions, establishing new properties like rationality and Lorentzian structure of the mixed Segre zeta function.

Findings

Proves the mixed Segre zeta function is rational.

Shows dependence only on the integral closure of ideals.

Establishes Lorentzian property of the homogenized numerator.

Abstract

We introduce and study the mixed Segre zeta function of a sequence of homogeneous ideals in a polynomial ring. This function is a power series encoding information about the mixed Segre classes obtained by extending the ideals to projective spaces of arbitrarily large dimension. Our work generalizes and unifies results by Kleiman and Thorup on mixed Segre classes and by Aluffi on Segre zeta functions. We prove that this power series is rational, with poles corresponding to the degrees of the generators of the ideals. We also show that the mixed Segre zeta function only depends on the integral closure of the ideals. Finally, we prove that the homogenization of the numerator of a modification of the mixed Segre zeta function is denormalized Lorentzian in the sense of Br\"and\'en and Huh.