On a Conjecture about the Number of Solutions to Linear Diophantine Equations with a Positive Integer Parameter

TL;DR

This paper investigates the number of solutions to parameterized linear Diophantine equations, conjecturing that this count is eventually a quasi-polynomial function of the parameter, and verifies this in specific cases.

Contribution

It proposes a conjecture that the solution count is a quasi-polynomial for large parameters and confirms this in several instances.

Findings

Conjecture that solution counts are eventually quasi-polynomials.

Verification of the conjecture in specific cases.

Provides evidence supporting the quasi-polynomial nature of solutions.

Abstract

Let A(n) be a $k\times s$ matrix and $m(n)$ be a $k$ dimensional vector, where all entries of A(n) and $m(n)$ are integer-valued polynomials in $n$. Suppose that $$t(m(n)|A(n))=#\{x\in\mathbb{Z}_{+}^{s}\mid A(n)x=m(n)\}$$ is finite for each $n\in \mathbb{N}$, where $Z_+$ is the set of nonnegative integers. This paper conjectures that $t(m(n)|A(n))$ is an integer-valued quasi-polynomial in $n$ for $n$ sufficiently large and verifies the conjecture in several cases.