# 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.

## Repository Citation

Roune, B.H., and K. Woods. 2015. "The Parametric Frobenius Problem." Electronic Journal of Combinatorics 22(2): P2.36.

## Publisher

Electronic Journal of Combinatorics

## Publication Date

5-22-2015

## Publication Title

Electronic Journal of Combinatorics

## Department

Mathematics

## Document Type

Article

## DOI

https://dx.doi.org/10.37236/5112

## Keywords

Quasi-polynomials

## Language

English

## Format

text