On Hilbert’s Construction of Positive Polynomials

In 1888, Hilbert described how to find real polynomials 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. 1. History and Overview A real polynomial f(x1, . . . , xn) is psd or positive if f(a) ≥ 0 for all a ∈ R; it is sos or a sum of squares if there exist real polynomials hj so that f = ∑ hj . For forms, we follow the notation of [4] and use Pn,m to denote the cone of real psd forms of even degree m in n variables, Σn,m to denote its subcone of sos forms and let ∆n,m = Pn,m r Σn,m. The Fundamental Theorem of Algebra implies that ∆2,m = ∅; ∆n,2 = ∅ follows from the diagonalization of psd quadratic forms. The first suggestion that a psd form might not be sos was made by Minkowski in the oral defense of his 1885 doctoral dissertation: Minkowski proposed the thesis that not every psd form is sos. Hilbert was one of his official “opponents” and remarked that Minkowski’s arguments had convinced him that this thesis should be true for ternary forms. (See [14], [15] and [24].) Three years later, in a single remarkable paper, Hilbert [11] resolved the question. He first showed that F ∈ P3,4 is a sum of three squares of quadratic forms; see [23] and [26] for recent expositions and [17, 18] for another approach. Hilbert then described a construction of forms in ∆3,6 and ∆4,4; after multiplying these by powers of linear forms if necessary, it follows that ∆n,m 6= ∅ if n ≥ 3 and m ≥ 6 or n ≥ 4 and m ≥ 4. The goal of this paper is to isolate the underlying mechanism of Hilbert’s construction, show that it applies to situations more general than those in [11], and use it to produce many new examples. In [11], Hilbert first worked with polynomials in two variables, which homogenize to ternary forms. Suppose f1(x, y) and f2(x, y) are two relatively prime real cubic polynomials with nine distinct real common zeros – {πi}, indexed arbitrarily – so that Date: July 14, 2007. 1991 Mathematics Subject Classification. Primary: 11E20, 11E25, 12D99, 14H50, 14N15. This material is based in part upon work of the author, supported by the USAF under DARPA/AFOSR MURI Award F49620-02-1-0325. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of these agencies.