Laurent polynomials and Eulerian numbers

Daniel Erman, Gregory G. Smith, Anthony Várilly-Alvarado

Published 2009-08-18, updated 2010-01-24Version 2

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.