Quasipolynomial behavior via constructibility in multigraded algebra

TL;DR

This paper demonstrates that constructible families of multigraded modules exhibit piecewise quasipolynomial growth in various algebraic invariants, linking Presburger counting functions with tame persistent homology.

Contribution

It establishes the piecewise quasipolynomial behavior of algebraic invariants for constructible modules over semigroup rings using Presburger counting and tame persistent homology.

Findings

Growth of local cohomology lengths is piecewise quasipolynomial.

Betti numbers and associated primes exhibit quasipolynomial growth.

Regularity and depth have quasilinear growth patterns.

Abstract

Piecewise quasipolynomial growth of Presburger counting functions combines with tame persistent homology module theory to conclude piecewise quasipolynomial behavior of constructible families of finely graded modules over constructible commutative semigroup rings. Functorial preservation of constructibility for families under local cohomology, $\operatorname{Tor}$, and $\operatorname{Ext}$ yield piecewise quasipolynomial, quasilinear, or quasiconstant growth statements for length of local cohomology, $a$-invariants, regularity, depth; length of $\operatorname{Tor}$ and Betti numbers; length of $\operatorname{Ext}$ and Bass numbers; associated primes via $v$-invariants; and extended degrees, including the usual degree, Hilbert-Samuel multiplicity, arithmetic degree, and homological degree.