2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134434In 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.Algebraic GeometryNumber Theory11E20, 11E25, 12D99, 14H50, 14N15text