Package alggeozd

This page describes the alggeozd package. The package contains various functions for checking algebraic and geometric properties of zero-dimensional affine K-algebra/schemes and related computations. For a complete list of functions, see Category:Package alggeozd.

Alg-Geo Properties

Let $K$ be a field,

Package Discription

The zerodim package provides functions for computing the introduced invariants of zero-dimensional schemes. In the graded case the package also provides functions for computations of the Hilbert functions of these invariants. Alias of the package is ZD.

List of main functions

MinQuotIdeal

MinQuotIdeal(P, I, J): computes a min. homog. system
          of generators of homog. ideal (I+J)/I.
     input: P=K[x[1..N]], I and J homog. ideals of P
     output: list of polys

Example for computations

Now let us apply the zerodim package to some concrete examples. Recall that the alias of the package is ZD, and so to call a function from this package in computation one uses ZD.functions-name.

Consider the first example, where X is the scheme defined by the homogeneous ideal Ix.

Use P ::= QQ[X[0..2]];
Ix := ideal(X[0]*X[1] -X[1]^2, X[1]^2*X[2] -X[1]*X[2]^2, 2*X[0]^2*X[2] -3*X[0]*X[2]^2 +X[2]^3);

Then we calculate the differents of X as follows: