Compact Formulae in Sparse Elimination

Ioannis Emiris 1
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , UoA - University of Athens
Abstract : It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-B ́ezout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.
Type de document :
Communication dans un congrès
International Symposium on Symbolic and Algebraic Computation (ISSAC '16), Jul 2016, Waterloo, Canada. Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC '16). <10.1145/2930889.2930943>
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https://hal.inria.fr/hal-01401132
Contributeur : Ioannis Emiris <>
Soumis le : mardi 22 novembre 2016 - 23:09:18
Dernière modification le : mercredi 23 novembre 2016 - 12:54:21

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Ioannis Emiris. Compact Formulae in Sparse Elimination. International Symposium on Symbolic and Algebraic Computation (ISSAC '16), Jul 2016, Waterloo, Canada. Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC '16). <10.1145/2930889.2930943>. <hal-01401132>

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