A Package for Computing Implicit Equations of Parametrizations from Toric Surfaces

In this paper we present an algorithm for computing a matrix representation for a surface in P3 parametrized over a 2-dimensional toric variety T embedded into a projective space. This algorithm follows the ideas of [BDD09] and [Bot11] and it is implemented in Macaulay2 [GS]. The authors show in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Bot11] that this can be generalized to the case where the base locus is not necessarily a locally complete intersection, but locally an almost complete intersection. The key point consists in exploiting the sparse structure of the parametrization, which allows us to avoid non almost complete intersection points and to obtain significantly smaller matrices than in the homogeneous case. This implementation contributes to computing implicit equations for rational surfaces in P3, avoiding costly Gröbner Bases methods and further, it permits to obtain matrix representations of such surfaces, which is better adapt to practical problems.