The parametric Frobenius problem (Q2346475)

Summary: Given relatively prime positive integers \(a_1,...,a_n\), 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 \(f:\mathbb{Z}_+\to\mathbb{Z}\) is a quasi-polynomial if there exists a period \(m\) and polynomials \(f_0,...,f_{m-1}\) such that \(f(t)=f_{t\mathrm{mod} m}(t)\) 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.