Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms.
Klerk, Etienne de.
Pasechnik, Dmitrii V.
Date of Issue2003
School of Physical and Mathematical Sciences
A form p on Rn (homogeneous n-variate polynomial) is called positive semidefinite (p.s.d.) if it is nonnegative on Rn. 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 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.
DRNTU::Science::Mathematics::Discrete mathematics::Theory of computation
European journal of operational research
© 2003 Elsevier. This is the author created version of a work that has been peer reviewed and accepted for publication by European Journal of Operational Research, Elsevier. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: DOI [http://dx.doi.org/10.1016/j.ejor.2003.08.014].