The Parametric Frobenius Problem

Given relatively prime positive integers a1, . . . , an, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the ai. We examine the parametric version of this problem: given ai = ai(t) as functions of t, compute the Frobenius number as a function of t. A function f : Z+ → Z is a quasi-polynomial if there exists a period m and polynomials f0, . . . , fm−1 such that f(t) = ft mod m(t) for all t. We conjecture that, if the ai(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 ai(t) are linear functions, and also prove it in the case where n (the number of generators) is at most 3.