On Hilbert's construction of positive polynomials

TL;DR

This paper revisits Hilbert's 1888 construction of non-negative polynomials that are not sums of squares, generalizing the method and providing explicit examples that were previously unknown.

Contribution

The paper generalizes Hilbert's original construction and offers numerous explicit examples of non-negative polynomials not expressible as sums of squares.

Findings

Many explicit examples of such polynomials are provided.

The generalization broadens the class of known non-negative polynomials.

Hilbert's construction is shown to be more versatile than previously understood.

Abstract

In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present many such polynomials.