Laurent polynomials and Eulerian numbers

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.

迪斯特马特（Duistermaat）和范德卡伦（van der Kallen）表明，不存在所有幂次都有零常数项的非平凡复洛朗多项式。受此启发，斯图姆费尔斯（Sturmfels）提出了两个问题：一般的洛朗多项式的常数项是否构成一个正则序列？如果是，那么相关的零维理想的次数是多少？在本文中，我们证明了欧拉数为第二个问题提供了答案。证明涉及从环面几何的角度重新解释该问题。