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

E De Klerk (E.deKlerkits.tudelft.nl)
D.V. Pasechnik (D.Pasechnikits.tudelft.nl)

Abstract: A form p on R^n (homogeneous n-variate polynomial) is called positive semidefinite (psd) 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 (later proven by Artin) is that a form p is psd if and only if it can be decomposed 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 psd ternary form p of degree 2m, 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.

Keywords: semidefinite programming, ternary forms, positive semidefinite forms

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Global Optimization (Theory )

Citation: Preprint, TU Delft, Mekelweg 4, 2628 CD, Delft, The Netherlands, 2001

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Entry Submitted: 11/06/2001
Entry Accepted: 11/06/2001
Entry Last Modified: 11/06/2001

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