Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms
In: European Journal of Operational Research, Jg. 157 (2004-08-16), Heft 1, S. 39-45
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Zugriff:
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. [Copyright &y& Elsevier]
Titel: |
Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms
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Autor/in / Beteiligte Person: | de Klerk, Etienne ; Pasechnik, Dmitrii V. |
Zeitschrift: | European Journal of Operational Research, Jg. 157 (2004-08-16), Heft 1, S. 39-45 |
Veröffentlichung: | 2004 |
Medientyp: | academicJournal |
ISSN: | 0377-2217 (print) |
DOI: | 10.1016/j.ejor.2003.08.014 |
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