Asymptotic Hilbert Polynomial and a bound for Waldschmidt constants

TL;DR

This paper introduces a new upper bound for Waldschmidt constants of a broad class of ideals, utilizing roots of derivatives of polynomials linked to asymptotic Hilbert polynomials, extending previous results.

Contribution

It generalizes prior bounds on Waldschmidt constants by connecting them to roots of derivatives of polynomials related to asymptotic Hilbert polynomials.

Findings

Provides a new upper bound for Waldschmidt constants.

Extends previous bounds to a wider class of ideals.

Uses roots of derivatives of associated polynomials for the bound.

Abstract

In the paper we give an upper bound for the Waldschmidt constants of the wide class of ideals. This generalizes the result obtained by Dumnicki, Harbourne, Szemberg and Tutaj-Gasinska, Adv. Math. 2014. Our bound is given by a root of a suitable derivative of a certain polynomial associated with the asymptotic Hilbert polynomial.