*MatrixRepToric* is a package for computing implicit equations of toric rational surfaces embedded in a projective space, by means of approximation complexes.

We provide a method for computing a matrix representation (see representationMatrix) and the implicit equation (see implicitEq ) by means of the method developed in [BDD09] and [Bot10]. As it is probably the most interesting case from a practical point of view, we restrict our computations to parametrizations from toric surfaces embedded in some projective space. This implementation allows to compute small examples for the better understanding of the theory developed in [BDD09] and [Bot10].

The aim of the present package is to provide an algorithm for computing a matrix representation for a surface in P^{3} parametrized over a 2-dimensional toric variety T embedded into a projective space. This algorithm follows the ideas of [BDD09] and [Bot10] 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 [Bot10] 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 Groebner Bases methods and further, it permits to obtain matrix representations of such surfaces, which is better adapt to practical problems.

References:

[BDD10] Nicolas Botbol, Marc Dohm, and Manuel Dubinsky. A package for computing implicit equations of parametrizations from toric surfaces, 2010. http://mate.dm.uba.ar/~nbotbol/pdfs/MatrixRepToric.pdf

[BDD09] Nicolas Botbol, Alicia Dickenstein, and Marc Dohm. Matrix representations for toric parametrizations. Comput. Aided Geom. Design, 26(7):757–771, 2009.

[Bot10] Nicolas Botbol. Compactifications of rational maps and the implicit equations of their images. J. Pure and Applied Algebra. 215(5):1053–1068, 2011.

[BC05] Laurent Buse and Marc Chardin. Implicitizing rational hypersurfaces using approximation complexes. J. Symbolic Comput, 40(4-5):1150–1168, 2005.

[BCJ09] Laurent Buse, Marc Chardin, and Jean-Pierre Jouanolou. Torsion of the symmetric algebra and implicitization. Proc. Amer. Math. Soc, 137(6):1855–1865, 2009.

[BCS10] Laurent Buse, Marc Chardin, and Aron Simis. Elimination and nonlinear equations of rees algebras. Journal of Algebra, 324(6):1314 – 1333, 2010.

[BD07] Laurent Buse and Marc Dohm. Implicitization of bihomogeneous parametrizations of algebraic surfaces via linear syzygies. In ISSAC 2007, pages 69–76. ACM, New York, 2007.

[BJ03] Laurent Buse and Jean-Pierre Jouanolou. On the closed image of a rational map and the implicitization problem. J. Algebra, 265(1):312–357, 2003.

[Cha06] Marc Chardin. Implicitization using approximation complexes. In Algebraic geometry and geometric modeling, Math. Vis, pages 23–35. Springer, Berlin, 2006.

[CLS11] David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties. Providence, RI: American Mathematical Society (AMS), 2011.

[Ful93] William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry.

[GKZ94] Israel M Gel′fand, Mikhail M Kapranov, and Aandrei V Zelevinsky. Discriminants, resultants, and multidimen- sional determinants. Mathematics: Theory & Applications. Birkh ̈auser Boston Inc, Boston, MA, 1994.

[KD06] Amit Khetan and Carlos D’Andrea. Implicitization of rational surfaces using toric varieties. J. Algebra, 303(2):543– 565, 2006.

- Marc Dohm
- Manuel Dubinsky

- Types
- ToricEmbedding -- Embedding of a variety with ambient space A^2 in an n-dimensional projective espace.

- Functions and commands
- degreeImplicitEq -- computes the degree of det((Z)_nu)
- implicitEq -- computes the gcd of the right-most map of the Z-complex in degree nu)
- isGoodDegree -- verifies if the Euler Chrasteristric of the Z-complex is zero in the given degree
- maxMinor -- Returns a maximal minor of the matrix of full rank.
- newToricEmbedding -- constructs a ToricEmbedding
- polynomialsToPolytope -- Return the convexHull of the union of the Newton polytope of all the polynomial in the list
- representationMatrix -- computes the right-most map of the Z-complex in degree nu)
- teGetPolytope -- returns the Polytope associated to the ToricEmbedding
- teLatticePotintsFromHomothethicPolytope -- Returns the list of coordinates of N based on the smallest homothety of P containing N.
- teToricRationalMap -- Computes the rational map G defined by {f0,f1,f2,f3} over the toric coordinate ring associated to a polytope P
- teToricRing -- Returns the coordinate ring of the toric variety

- Methods
- net(ToricEmbedding) -- Defines the new type ToricEmbedding
- teGetPolytope(ToricEmbedding), see teGetPolytope -- returns the Polytope associated to the ToricEmbedding
- teToricRing(ToricEmbedding), see teToricRing -- Returns the coordinate ring of the toric variety