Laurent polynomials and Eulerian numbers

Authors:
Daniel Erman;Gregory G. Smith;Anthony Várilly-Alvarado
Affiliations:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA;Department of Mathematics & Statistics, Queen's University, Kingston, ON, K7L 3N6, Canada;Department of Mathematics, MS 136, Rice University, 6100 South Main Street, Houston, TX 77005-1892, USA
Venue:
Journal of Combinatorial Theory Series A
Year:
2011

Citing 3
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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 poses 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.