Matrix representations for toric parametrizations

Abstract

In this paper we show that a surface in P 3 parametrized over a 2-dimensional toric variety T 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 result over P 2 established in [BJ03] and [BC05]. 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. We also treat the important case T = P 1 × P 1 in detail and give numerous examples.

Implicitization of rational hypersurfaces via linear syzygies: A practical overview

“…Precise hypotheses and proof of Theorem 3.3. In this subsection we detail the precise hypotheses that ensure the validity of Theorem 3.3 and we prove it, based on results in [BDD09]. We first need to recall a few standard definitions from commutative algebra.…”

“…We refer to [Cox03b,Ful93,CLS11] and [GKZ94, Ch.5&6] for the general notions, and to [KD06,§2], [BDD09,Bot11a] for applications to the implicitization problem. Any reader only interested in the application of Algorithm 3.1 or its bihomogeneous (toric) refinement given in Algorithm 3.6 can skip this section.…”

“…We point out in Remark 4.7 the possible outcomes. The general algorithm can be refined using Theorem 11 in [BDD09].…”

“…This is why we postpone the detail of Hypotheses 4.4 and 4.8 until Section 4. Our approach is an inhomogeneous translation of the basic general algorithm for the sparse case in [BDD09], which were inspired by the methods [BJ03] for classical homogeneous polynomials.…”

“…Goldman et al studied the cases of planar and space curves [JG09,HWJG10,JWG10]. A generalization of the linear syzygy method when the support of the input polynomials is a square (that is, bihomogeneous of degree (d, d)) was proposed by Busé and Dohm [BD07], and for any polygon by Botbol, Dickenstein and Dohm [BDD09], and Botbol [Bot09,Bot11a,Bot11b]. This method is particularly adapted when the polynomials defining the parametrization are sparse, which is often the case.…”