Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms

Abstract

A form p on R n (homogeneous n-variate polynomial) is called positive semidefinite (p.s.d.) if it is nonnegative on R n . In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert [Bull. Amer. Math. Soc. (N.S.), 37 (4) (2000) 407] (later proven by Artin [The Collected Papers of Emil Artin, Addison-Wesley Publishing Co., Inc., Reading, MA, London, 1965]) is that a form p is p.s.d. if and only if it can be decomposed into a sum of squares of rational functions. In this paper we give an algorithm to compute such a decomposition for ternary forms ( n=3). This algorithm involves the solution of a series of systems of linear matrix inequalities (LMI's). In particular, for a given p.s.d. ternary form p of degree 2 m, we show that the abovementioned decomposition can be computed by solving at most m/4 systems of LMI's of dimensions polynomial in m. The underlying methodology is largely inspired by the original proof of Hilbert, who had been able to prove his conjecture for the case of ternary forms.