In the present thesis, we try to compare the classical boolean complexity with the algebraic complexity, by studying problems related to polynomials. We consider the algebraic models from Valiant and from Blum, Shub and Smale (BSS). To study the algebraic complexity classes, one can start from results and open questions from the boolean case, and look at their translation in the algebraic context. The comparison of the results obtained in the two settings will then boost our understanding of both complexity theories. The first part follows this framework. By considering a polynomial canonically associated to a boolean formula, we get a link between boolean complexity issues on the formula and algebraic complexity problems on the polynomial. We studied the complexity of computing the polynomial in Valiant's model, as a function of the complexity of the boolean formula. We found algebraic counterparts to some boolean results. Along the way, we could also use some algebraic methods to improve boolean results, in particular by getting better counting reductions. Another motivation for algebraic models of computation is to offer an elegant framework to the study of numerical algorithms. The second part of this thesis follows this approach. We started from new algorithms for the search of approximate zeros of complex systems of n polynomials in n variables. Up to now, those were BSS machine algorithms. We studied the implementation of these algorithms on digital computers, and propose an algorithm using floating arithmetic for this problem.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00665782 |
| Date | 29 November 2011 |
| Creators | Briquel, Irénée |
| Publisher | Ecole normale supérieure de lyon - ENS LYON |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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