Laurent polynomials and Eulerian numbers

TL;DR

This paper explores the relationship between Laurent polynomials and Eulerian numbers, using toric geometry to answer questions about the regularity and degree of ideals formed by constant terms.

Contribution

It demonstrates that Eulerian numbers determine the degree of the ideal generated by constant terms of a generic Laurent polynomial, linking combinatorics and algebraic geometry.

Findings

Eulerian numbers give the degree of the ideal of constant terms

The problem is interpreted through toric geometry

Answers a question about regular sequences of Laurent polynomial constants

Abstract

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels posed two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.