Logarithmic Discriminants of Hyperplane Arrangements

TL;DR

This paper investigates the properties of the logarithmic discriminant associated with hyperplane arrangements, linking it to the Hurwitz form of reciprocal linear spaces, with implications for critical point analysis in physics and statistics.

Contribution

It introduces a novel study of the logarithmic discriminant's properties and its connection to the Hurwitz form of reciprocal linear spaces, advancing understanding in algebraic geometry.

Findings

Characterization of the logarithmic discriminant in hyperplane arrangements

Connection established between the discriminant and Hurwitz form of reciprocal linear spaces

Insights into critical point degeneracies in affine-linear function products

Abstract

A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space.