On Exact Polya, Hilbert-Artin and Putinar's Representations

Victor Magron 1 Mohab Safey El Din 2
2 PolSys - Polynomial Systems
Inria de Paris, LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : We consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of non-negative multivariate polynomials, relying on semidefinite programming (SDP) solvers. We provide a hybrid numeric-symbolic algorithm computing exact rational SOS decompositions for polynomials lying in the interior of the SOS cone. It computes an approximate SOS decomposition for a perturbation of the input polynomial with an arbitrary-precision SDP solver. An exact SOS decomposition is obtained thanks to the perturbation terms. We prove that bit complexity estimates on output size and runtime are both polynomial in the degree of the input polynomial and simply exponential in the number of variables. Next, we apply this algorithm to compute exact Polya, Hilbert-Artin's representation and Putinar's representations respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. We also report on practical experiments done with the implementation of these algorithms and existing alternatives such as the critical point method and cylindrical algebraic decomposition.
Type de document :
Pré-publication, Document de travail
29 pages, 4 tables, extended version of the paper from ISSAC'18 conference (available at arXiv::1.. 2018
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https://hal.archives-ouvertes.fr/hal-01935727
Contributeur : Victor Magron <>
Soumis le : mardi 27 novembre 2018 - 09:13:09
Dernière modification le : lundi 3 décembre 2018 - 11:02:49

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  • HAL Id : hal-01935727, version 1
  • ARXIV : 1811.10062

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Victor Magron, Mohab Safey El Din. On Exact Polya, Hilbert-Artin and Putinar's Representations. 29 pages, 4 tables, extended version of the paper from ISSAC'18 conference (available at arXiv::1.. 2018. 〈hal-01935727〉

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