Implicitization of rational (hyper)surfaces

A. DICKENSTEIN

Sydney, Australia

Congreso; 1st Prima Congress; 2009

University of New South Wales, Australia

We shall present some advances on two different approaches for the basic problem of turning a parametrization of a rational projective hypersurface H into an implicit equation, based on the structure of the polynomials defining the parametrization. The first approach is to compute a representation matrix for H, that is, a matrix M of full rank with polynomial entries, and such that a given point p lies in H if and only if the rank of M evaluated at p is not maximum (as suggested by L. Busé and M. Dohm). In common work with N. Botbol and M. Dohm, we show that a surface in P^3 parametrized over a 2-dimensional toric variety can be represented by a matrix of linear syzygies, if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding results over P^2 established by L. Busé, M. Chardin and J.-P. Jouanolou (based on the approximation complexes introduced by A. Simis and W. Vasconcelos).  Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. The second approach, following common work with B. Mourrain, extends methods from tropical implicitization developed by B. Sturmfels, J. Tevelev and J. Yu, to predict the Newton  polytope of an implicit equation of H.