# Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée

Abstract : When dealing with multivariate real polynomials, two natural questions arise : decide if a given polynomial is positive and compute its minimum.

This thesis is devoted to those problems when the study is led on a simplex of $\R^k$. The main tool is the Bernstein basis, more suited in this case than the traditional monomial basis. In particular, its positivity and bounding properties are essential.

We first derive an algorithm deciding if a given polynomial $f$ is positive on a simplex $V$ of $\R^k$, and giving, if need be, an expression of $f$ that makes its positivity trivial : a so-called certificate of positivity.

We also derive an algorithm for minimizing a polynomial $f$ over a simplex $V$. Both algorithms are certified, and their complexity is studied in this thesis.
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https://tel.archives-ouvertes.fr/tel-00349444
Contributor : Richard Leroy <>
Submitted on : Tuesday, December 30, 2008 - 6:42:35 PM
Last modification on : Thursday, January 7, 2021 - 4:25:28 PM
Long-term archiving on: : Thursday, October 11, 2012 - 2:55:52 PM

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• HAL Id : tel-00349444, version 1

### Citation

Richard Leroy. Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée. Mathématiques [math]. Université Rennes 1, 2008. Français. ⟨tel-00349444⟩

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