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Rationality of the multivariate growth series for algebraic sets of virtually abelian groups

Yusuke Nakamura

TL;DR

This work proves that the multivariate relative growth series for algebraic sets in finitely generated virtually abelian groups are rational functions, resolving a conjecture of Evetts and Levine. The authors recast the problem in a periodic-graph framework, construct and decompose a graded growth set $B$ into finitely generated monoid modules $X_S$ with corresponding finitely generated submonoids $M_S$, and apply Hilbert-series rationality to obtain rationality of the multivariate generating functions. Extending EL22's coset-polyhedral decomposition, they show algebraic sets $U \subset G^d$ decompose into pieces each contributing a rational multivariate growth series, thereby proving rationality for all such $U$. The results bridge growth series, monoid/module theory, and polyhedral decompositions, offering a robust method with potential further applications in Ehrhart-type enumeration for virtually abelian groups.

Abstract

We prove the rationality of the multivariate relative growth series for algebraic sets of virtually abelian groups, which had been conjectured by Evetts and Levine.

Rationality of the multivariate growth series for algebraic sets of virtually abelian groups

TL;DR

This work proves that the multivariate relative growth series for algebraic sets in finitely generated virtually abelian groups are rational functions, resolving a conjecture of Evetts and Levine. The authors recast the problem in a periodic-graph framework, construct and decompose a graded growth set into finitely generated monoid modules with corresponding finitely generated submonoids , and apply Hilbert-series rationality to obtain rationality of the multivariate generating functions. Extending EL22's coset-polyhedral decomposition, they show algebraic sets decompose into pieces each contributing a rational multivariate growth series, thereby proving rationality for all such . The results bridge growth series, monoid/module theory, and polyhedral decompositions, offering a robust method with potential further applications in Ehrhart-type enumeration for virtually abelian groups.

Abstract

We prove the rationality of the multivariate relative growth series for algebraic sets of virtually abelian groups, which had been conjectured by Evetts and Levine.
Paper Structure (10 sections, 13 theorems, 31 equations)

This paper contains 10 sections, 13 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Growth series of periodic graphs
  4. Growth series of groups
  5. Some facts in commutative monoid theory
  6. Rationality of the growth series of periodic graphs
  7. Rationality of the multivariate growth series of algebraic sets of virtually abelian groups
  8. Algebraic structure of the graded growth set
  9. Notations and Preliminaries
  10. Proof of Theorem \ref{['thm:NSMN_decomp']}

Key Result

Theorem 1.1

Let $G$ be a finitely generated virtually abelian group, $S \subset G$ a finite subset, and $\omega$ a positive integer weight. Then, for any algebraic subset $U \subset G^d$ of $G$, its multivariate growth series $\mathbb{S}_{Y, S, \omega}({\bf z})$ is a rational function.

Theorems & Definitions (30)

  • Theorem 1.1: $=$ Theorem \ref{['thm:main']}
  • Definition 2.2: cf. IN1
  • Theorem 2.3: NSMN21*Theorem 2.2
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Benson Ben83
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9: cf. BG09*Theorem 6.37
  • Theorem 3.1: NSMN21
  • ...and 20 more