David A. Cox

Ideals, Varieties, and Algorithms

(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In Maple, a monomial ordering is called a monomial order. The monomial orderings lex, grlex, and grevlex from Chapter 2 are easy to use in Maple. Lex order is called plex (for “pure lexicographic”), grlex order is called grlex, and grevlex order is called tdeg (for “total degree”). Be careful not to confuse tdeg with grlex. Since a monomial order depends also on how the variables are ordered, Maple needs to know both the monomial order you want (plex, grlex or tdeg) and a list of variables. For example, to tell Maple to use lex order with variables x > y > z, you would need to input plex(x,y,z). The Groebner package also knows some elimination orders, as defined in Exercise 5 of Chapter 3, §1. To eliminate the first k variables from x1, . . . , xn, one can use the monomial order lexdeg([x 1,. . .,x k],[x {k+1},. . . ,x n]) (remember that Maple encloses a list inside brackets [. . .]). This order is the elimination order of Bayer and Stillman described in Exercise 6 of Chapter 3, §1. The Maple documentation for the Groebner package also describes how to use certain weighted orders, and we will explain below how matrix orders give us many more monomial orderings. The most commonly used commands in the Groebner package are NormalForm, for doing the division algorithm, and Basis, for computing a Groebner basis. NormalForm has the following syntax:

On the Hodge structure of projective hypersurfaces in toric varieties

This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a

The moving line ideal basis of planar rational curves

d

The Arithmetic-Geometric Mean of Gauss

-dimensional projective simplicial toric variety

On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points

P

The μ -basis and implicitization of a rational parametric surface

and an ample hypersurface

Residues in toric varieties

X

Toric Varieties and Toric Resolutions

defined by an polynomial

Introduction to Fermat's last theorem

f

The moving curve ideal and the Rees algebra

in the homogeneous coordinate ring