The Parametric Frobenius Problem

TL;DR

This paper investigates the parametric Frobenius problem, proving that the Frobenius number behaves as a quasi-polynomial for large t when the generators are linear functions or when there are at most three generators.

Contribution

It proves the conjecture that the Frobenius number is a quasi-polynomial in t for linear generators and for up to three generators, advancing understanding of the parametric Frobenius problem.

Findings

Frobenius number is a quasi-polynomial for linear generators.

Confirmed the conjecture for cases with up to three generators.

Established the behavior of Frobenius number as t grows large.

Abstract

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 is a quasi-polynomial if there exists a period m and polynomials f_0,...,f_{m-1} such that f(t)=f_{t mod m}(t) for all positive integers 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.