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

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

2007-09-30

arxiv.org/abs/0710.0177v1

Mathematics

Combinatorics

14 pages

Scientific paper

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.

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