Implicit polynomial support optimized for sparseness

Authors:
Ioannis Z. Emiris;Ilias S. Kotsireas
Affiliations:
Department of Informatics & Telecommunications, National University of Athens, Panepistimiopolis, Greece and INRIA Sophia, Antipolis, France;Department of Computing, Wilfrid Laurier University, Waterloo, ON, Canada and Ontario Research Centre for Computer Algebra, University of Western Ontario, London, ON, Canada
Venue:
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartIII
Year:
2003

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Abstract

We propose the use of various tools from algebraic geometry, with an emphasis on toric (or sparse) elimination theory, in order to predict the support of the implicit equation of a parametric hypersurface. The problem of implicitization lies at the heart of several algorithms in geometric modeling and computer-aided design, two of which (based on interpolation) are immediately improved by our contribution. We believe that other methods of implicitization shall be able to benefit from our work. More specifically, we use information on the support of the toric resultant, and degree bounds, formulated in terms of the mixed volume of Newton polytopes. The computed support of the implicit equation depends on the sparseness of the parametric expressions and is much tighter than the one predicted by degree arguments. Our Maple implementation illustrates many cases in which we obtain the exact support. In addition, it is possible to specify certain coefficients of the implicit equation.