On the effective Pourchet's Theorem

TL;DR

The paper addresses the problem of algorithmically representing positive univariate polynomials with rational coefficients as sums of squares of rational polynomials, aiming for an effective five-square decomposition in line with Pourchet's bound. It refines the 2-adic Newton polygon approach and employs p-adic Hensel lifting to produce new five-square decompositions and to identify limitations of prior MKV23 conjectures. A key contribution is a counterexample family $f_{k,N}$ showing that Algorithm 9 from MKV23 does not terminate, clarifying the role of $2$-adic valuation conditions, and proposing a modified algorithm that works for broader cases, including when the degree is a multiple of 4. The work advances the constructive, algorithmic aspect of Pourchet-type decompositions and outlines remaining gaps toward a complete effective theory for all positive rational polynomials.

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.

Theorems & Definitions (26)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1: jac35jac36
• Proposition 2.2
• proof
• Remark 2.3