Exact matrix formula for the unmixed resultant in three variables

We give the first exact determinantal formula for the resultant of an unmixed sparse system of four Laurent polynomials in three variables with arbitrary support. This follows earlier work by the author on exact formulas for bivariate systems and also uses the exterior algebra techniques of Eisenbud and Schreyer. Along the way we will prove an interesting new vanishing theorem for the sheaf cohomology of divisors on toric varieties. This will allow us to describe some supports in four or more variables for which determinantal formulas for the resultant exist.Comment: 24 pages, 2 figures, Cohomology vanishing theorem generalized with new combinatorial proof. Can state some cases of exact resultant formulas in higher dimensio

The Resultant of an Unmixed Bivariate System

This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid Sylvester and B\'ezout type. We make use of the exterior algebra techniques of Eisenbud, Fl{\o}ystad, and Schreyer.Comment: 18 pages, 2 figure

Combinatorial construction of toric residues

The toric residue is a map depending on n+1 semi-ample divisors on a complete toric variety of dimension n. It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when n=2 and for any n when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier results when the divisors were all ample.Comment: 29 pages, 9 pstex figures, 1 large eps figure. New title, a few typos corrected, to appear in Ann. Inst. Fourie

Implicitization of rational surfaces using toric varieties

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of moving surfaces. Paper accepted and to appear in Journal of Algebr