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Pourchet proved in 1971 that every nonnegative univariate polynomial withrational coefficients is a sum of five or fewer squares. Nonetheless, there areno known algorithms for constructing such a decomposition. The sole purpose ofthe present paper is to present a set of algorithms that decompose a givennonnegative polynomial into a sum of six (five under some unproven conjectureor when allowing weights) squares of polynomials. Moreover, we prove that thebinary complexity can be expressed polynomially in terms of classicaloperations of computer algebra and algorithmic number theory.

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