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Factoring polynomials over function fields

Jose Felipe Voloch

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.

Factoring polynomials over function fields

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.

Paper Structure

This paper contains 10 sections, 7 theorems, 11 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The basic rings
  4. Gaussian elimination
  5. Hasse derivatives and Wronskians
  6. The main algorithm
  7. Running time estimates and finding a suitable place
  8. Other applications
  9. Arithmetic properties
  10. List decoding

Key Result

Lemma 2.1

If $u \in R_1$ satisfies $D^{(i)}(u)=0, 0 < i < q$, then in each local summand of $R_1$, $u$ is constant (in the above sense).

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1
  • ...and 8 more