Factoring polynomials over function fields

TL;DR

This paper presents a versatile algorithm for factoring polynomials over function fields in positive characteristic, enabling efficient root finding, irreducibility testing, and applications in algebraic settings.

Contribution

It introduces a general, flexible polynomial factorization algorithm over function fields with deterministic irreducibility testing in small characteristic.

Findings

Algorithm successfully factors polynomials over function fields.

Efficient root finding within specified subspaces.

Deterministic polynomial-time irreducibility test in small characteristic.

Abstract

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional restrictions, e.g., to find all roots that belong to a given finite dimensional k-subspace of K more efficiently. It also provides a deterministic polynomial time irreducibility test in small characteristic. We also discuss some applications.