Variation of Archimedean Zeta Function and $n/d$-Conjecture for Generic Multiplicities

TL;DR

This paper introduces the variation of Archimedean zeta functions and proves the $n/d$-conjecture for generic multiplicities, confirming the strong monodromy conjecture for hyperplane arrangements with such multiplicities.

Contribution

It defines the variation of Archimedean zeta functions and proves the $n/d$-conjecture for generic multiplicities, advancing understanding of monodromy in hyperplane arrangements.

Findings

$n/d$-conjecture holds for generic multiplicities

Strong monodromy conjecture confirmed for hyperplane arrangements with generic multiplicities

Introduction of variation of Archimedean zeta function

Abstract

For $f_1,...,f_r\in \mathbb C[z_1,...,z_n]\setminus \mathbb C$, we introduce the variation of archimedean zeta function. As an application, we show that the $n/d$-conjecture, proposed by Budur, Musta\c{t}\u{a}, and Teitler, holds for generic multiplicities. Consequently, strong monodromy conjecture holds for hyperplane arrangements with generic multiplicities as well.