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Habilitation à diriger des recherches

Méthodes d'élimination et applications

Abstract : This thesis of habilitation provides a systematic treatment of elimination algorithms that compute various zero decompositions for systems of multivariate polynomials. The central concepts are triangular sets and systems of different kinds, in terms of which the decompositions are represented. Some of the operations and results on multivariate polynomials which are used throughout the thesis are collected in the first chapter. Chaps. 2 to 5 are devoted to describing the algorithms of zero decomposition. We start by presenting algorithms that decompose arbitrary polynomial systems into triangular systems; the latter are not guaranteed to have zeros. These algorithm are modified in Chap. 3 by incorporating the projection process and GCD computation so that the computed triangular systems always have zeros. Then, we elaborate how to make use of polynomial factorization in order to compute triangular systems that are irreducible. Many of the algorithms and their underlying theories are proposed and developed by the author on the basis of the previous work of J.F. Ritt, W.-t. Wu, A. Seidenberg and J.M. Thomas. A brief review of some relevant algorithms including those based on resultants and Groebner bases is given in Chap. 5. Elimination methods play a special role in constructive algebraic geometry and polynomial ideal theory. Chap. 6 contains investigations on a few problems from these two areas. The last three chapters of the thesis discuss several selected applications of symbolic elimination methods such as solving algebraic systems and automated geometric reasoning.
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Habilitation à diriger des recherches
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Dongming Wang. Méthodes d'élimination et applications. Modélisation et simulation. Institut National Polytechnique de Grenoble - INPG, 1999. ⟨tel-00004862⟩

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