Implicitization of rational algebraic surfaces with syzygy-based methods

Marc Dohm 1, 2
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis, CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : The implicitization of a rational algebraic surface, i.e. the passage from a parametrization to an implicit representation, is a classical geometric problem. In this thesis we use the theory of syzygies to represent a surface implicitly by a matrix whose maximal-sized minors have the implicit equation of the surface as their greatest common divisor. In the first two chapters, we treat two special classes of surfaces for which it is always possible to construct a square representation matrix corresponding to the resultant of a μ-basis: ruled surfaces and canal surfaces. In the following chapters, the general case of rational surfaces parametrized over a two-dimensional toric variety is studied. We show that a representation matrix can be constructed only using linear syzygies and we give a simple and efficient algorithm for its computation.
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Marc Dohm. Implicitization of rational algebraic surfaces with syzygy-based methods. Mathematics [math]. Université Nice Sophia Antipolis, 2008. English. ⟨tel-00294484⟩

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