Integral points on singular toric varieties and cyclic normal polynomials

TL;DR

This paper develops a height zeta function formula for integral points on singular toric varieties, extending harmonic analysis methods to include cyclic quotient singularities, and applies it to count certain polynomials with Galois groups.

Contribution

It introduces a new approach to counting integral points on singular toric varieties using harmonic analysis, applicable to varieties with cyclic quotient singularities.

Findings

Derived a formula for the height zeta function on singular toric varieties.

Determined the asymptotic count of monic integral polynomials with specified Galois groups.

Extended harmonic analysis techniques to varieties with cyclic quotient singularities.

Abstract

We establish a formula for the height zeta function for integral points on a class of projective toric varieties. Our method builds on the harmonic analysis approach of Batyrev--Tschinkel for rational points and is applicable even when the toric variety has cyclic quotient singularities. As an application, we determine the leading term in the asymptotic number of monic integral polynomials of bounded height with linearly independent roots and a given cyclic Galois group.