The Parametric Frobenius Problem

Abstract

Given relatively prime positive integers $a_1,\ldots,a_n$, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the $a_i$. We examine the parametric version of this problem: given $a_i=a_i(t)$ as functions of $t$, compute the Frobenius number as a function of $t$. A function $f:\mathbb{Z}_+\rightarrow\mathbb{Z}$ is a quasi-polynomial if there exists a period $m$ and polynomials $f_0,\ldots,f_{m-1}$ such that $f(t)=f_{t\bmod m}(t)$ for all $t$. We conjecture that, if the $a_i(t)$ are polynomials (or quasi-polynomials) in $t$, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large $t$. We prove this in the case where the $a_i(t)$ are linear functions, and also prove it in the case where $n$ (the number of generators) is at most 3.