Quasipolynomial behavior via constructability in multigraded algebra
Hailong Dao, Ezra Miller, Jonathan Montaño, Christopher O'Neill, Kevin Woods
TL;DR
This work develops a unifying framework that couples Presburger arithmetic with tame, constructible multigraded modules to prove piecewise quasipolynomial growth for a wide class of families over constructible semigroup rings. It shows that key functors (local cohomology, Tor, Ext) preserve constructibility of families, enabling piecewise quasipolynomial behavior for invariants such as lengths, Betti/Bass numbers, and various degrees across parameter spaces indexed by Rees monoids. The approach blends logic, combinatorics, and homological algebra to extend classical noetherian growth results to nonnoetherian, multi-parameter settings, including ideals and their operations. The results provide a versatile toolkit for predicting asymptotic numerics in multigraded algebra and related areas, with potential algorithmic applications in computable Presburger frameworks and multiparameter persistent homology.
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.