Mixed-volume computation by dynamic lifting applied to polynomial system solving |
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Authors: | J Verschelde K Gatermann R Cools |
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Institution: | (1) Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3001 Heverlee, Belgium;(2) Konrad-Zuse-Zentrum für Informationstechnik Berlin, Heilbronner Str. 10, D-10711 Berlin-Wilmersdorf, Germany |
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Abstract: | The aim of this paper is to present a flexible approach for the efficient computation of the mixed volume of a tuple of polytopes.
In order to compute the mixed volume, a mixed subdivision of the tuple of polytopes is needed, which can be obtained by embedding
the polytopes in a higher-dimensional space, i.e., by lifting them. Dynamic lifting is opposed to the static approach. This
means that one considers one point at a time and only fixes the value of the lifting function when the point really influences
the mixed volume. Conservative lifting functions have been developed for this purpose. This provides us with a deterministic
manipulation of the lifting for computing mixed volumes, which rules out randomness conditions. Cost estimates for the algorithm
are given. The implications of dynamic lifting on polyhedral homotopy methods for the solution of polynomial systems are investigated
and applications are presented. |
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Keywords: | |
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