On a Frobenius problem for polynomials

TL;DR

This paper extends the Frobenius problem to polynomials over a field, providing solutions for the two-variable case, translating classical results, and offering an algorithm for large fields.

Contribution

It introduces a polynomial Frobenius problem framework, solves the two-variable case, and develops an algorithm for large fields, expanding classical number theory to polynomial rings.

Findings

The n=2 polynomial Frobenius problem is easily solvable.

Classical Frobenius results are adapted to polynomial rings.

An algorithm is provided for fields of sufficiently large size.

Abstract

We extend the famous diophantine Frobenius problem to the case of polynomials over a field $k$. Similar to the classical problem, we show that the $n=2$ case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over $\mathbb{Z}$ to $k[t]$ and give an algorithm to solve the Frobenius problem for polynomials over a field $k$ of sufficiently large size.