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

 dc.contributor.author Klerk, Etienne de. dc.contributor.author Pasechnik, Dmitrii V. dc.date.accessioned 2012-07-03T01:48:21Z dc.date.available 2012-07-03T01:48:21Z dc.date.copyright 2003 dc.date.issued 2012-07-03 dc.identifier.citation Klerk, E. D., & Pasechnik, D. V. (2003). Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms. European Journal of Operational Research, 157(1), 39–45. dc.identifier.uri http://hdl.handle.net/10220/8271 dc.description.abstract 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. dc.format.extent 13 p. dc.language.iso en dc.relation.ispartofseries European journal of operational research dc.rights © 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]. dc.subject DRNTU::Science::Mathematics::Discrete mathematics::Theory of computation. dc.title Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms. dc.type Journal Article dc.contributor.school School of Physical and Mathematical Sciences dc.identifier.doi http://dx.doi.org/10.1016/j.ejor.2003.08.014 dc.description.version Accepted version

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