Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares

TL;DR

This paper introduces algorithms to decompose nonnegative univariate polynomials into sums of five or six squares, providing complexity bounds and advancing computational methods for polynomial sum-of-squares representations.

Contribution

It presents the first algorithms for constructing such decompositions and analyzes their polynomial complexity, building on Pourchet's theorem.

Findings

Algorithms for sum-of-five-squares decomposition

Polynomial complexity bounds established

Decomposition into six squares with proven methods

Abstract

Pourchet proved in 1971 that every nonnegative univariate polynomial with rational coefficients is a sum of five or fewer squares. Nonetheless, there are no known algorithms for constructing such a decomposition. The sole purpose of the present paper is to present a set of algorithms that decompose a given nonnegative polynomial into a sum of six (five under some unproven conjecture or when allowing weights) squares of polynomials. Moreover, we prove that the binary complexity can be expressed polynomially in terms of classical operations of computer algebra and algorithmic number theory.