29 research outputs found
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
Bezoutians and Tate Resolutions
This paper gives an explicit construction of the Tate resolution of sheaves
arising from the d-fold Veronese embedding of P^n. Our description involves the
Bezoutian of n+1 homogenous forms of degree d in n+1 variables. We give
applications to duality theorems, including Koszul duality.Comment: 11 pages, LaTex2e using xypi
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
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
Computing in algebraic geometry and commutative algebra using Macaulay 2
AbstractWe present recent research of Eisenbud, Fløystad, Schreyer, and others, which was discovered with the help of experimentation with Macaulay 2. In this invited, expository paper, we start by considering the exterior algebra, and the computation of Gröbner bases. We then present, in an elementary manner, the explicit form of the Bernstein–Gelfand–Gelfand relationship between graded modules over the polynomial ring and complexes over the exterior algebra, that Eisenbud, Fløystad and Schreyer found. We present two applications of these techniques: cohomology of sheaves, and the construction of determinantal formulae for (powers of) Macaulay resultants. We show how to use Macaulay 2 to perform these computations
Compact Formulae in Sparse Elimination
International audienceIt has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects