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Cyclotomic generating functions, empty weighted complete intersections and positivity

Mona Gatzweiler, Fabián Levicán-Santibáñez, Atsuro Yoshida

Abstract

We give a sufficient combinatorial condition for the non-negativity of the coefficients of polynomial quotients of products of $q$-integers, also known as cyclotomic generating functions (CGFs). This slightly extends work by Iano-Fletcher, Pizzato, Sano and Tasin, who studied this condition as a criterion for quasismoothness of complete intersections in weighted projective spaces. As a consequence, we solve a problem by Billey and Swanson, prove most cases of an unpublished conjecture by Stanton and most cases of two conjectures by Gatzweiler and Krattenthaler. We also study sufficient conditions given by structural properties of the division lattice.

Cyclotomic generating functions, empty weighted complete intersections and positivity

Abstract

We give a sufficient combinatorial condition for the non-negativity of the coefficients of polynomial quotients of products of -integers, also known as cyclotomic generating functions (CGFs). This slightly extends work by Iano-Fletcher, Pizzato, Sano and Tasin, who studied this condition as a criterion for quasismoothness of complete intersections in weighted projective spaces. As a consequence, we solve a problem by Billey and Swanson, prove most cases of an unpublished conjecture by Stanton and most cases of two conjectures by Gatzweiler and Krattenthaler. We also study sufficient conditions given by structural properties of the division lattice.
Paper Structure (9 sections, 19 theorems, 45 equations)

This paper contains 9 sections, 19 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multisets
  4. Cyclotomic polynomials
  5. Cyclotomic generating functions (CGFs)
  6. The Frobenius coin problem
  7. Regular sequences
  8. The algebro-geometric point of view
  9. The lattice-theoretic and analytic points of view

Key Result

Proposition 1.4

Let $a_1, \ldots, a_n, b_1, \ldots, b_n \in \mathbb{Z}_+$. Let $A = [a_i]_{i = 1, \ldots, n}$ and $B = [b_i]_{i = 1, \ldots, n}$ be the corresponding multisets. Then, the following are equivalent:

Theorems & Definitions (36)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Proposition 1.4: Proposition \ref{['proposition:regular-sequence-iff-wci-iff-frobenius-coin-problems']}
  • Theorem 1.6: Theorem \ref{['theorem:main-theorem']}
  • Corollary 1.7: Corollary \ref{['corollary:conjecture-gk-upper-bound']}
  • Corollary 1.8: Corollary \ref{['corollary:conjecture-stanton-upper-bound']}
  • Proposition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • ...and 26 more