On the effective Pourchet's Theorem

TL;DR

This paper refines an algorithm using Hensel Lemma to express positive rational polynomials as sums of five squares, extending its applicability and providing new examples beyond previous methods.

Contribution

It improves the 2-adic Newton polygon algorithm for effective Pourchet's Theorem and extends its coverage to nearly all inputs.

Findings

Successfully expresses positive polynomials as sums of five squares.

Extends algorithm to cover almost all possible inputs.

Provides new examples beyond previous conjectural algorithms.

Abstract

With the aid of Hensel Lemma, we refine the 2-adic Newton polygon algorithm proposed by Magron, Koprowski, and Vaccon at ISSAC 2023 to express computationally a given positive univariate polynomial with rational coefficients as a sum of five squares of rational polynomials -the effective Pourchet's Theorem- and extend it to cover almost all the possible inputs. We also provide examples which are covered with our methods but cannot be detected by previous conjectural algorithms.