The Parametric Frobenius Problem

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 : Z(+) -> Z 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 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.

Publisher

Electronic Journal of Combinatorics

Publication Date

5-22-2015

Publication Title

Electronic Journal of Combinatorics

Department

Mathematics

Document Type

Article

Keywords

Quasi-polynomials

Language

English

Format

text

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