Documentation

groebner::dimension

Dimension of the affine variety generated by polynomials

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Syntax

groebner::dimension(polys, <order>)

Description

groebner::dimension(polys) computes the dimension of the affine variety generated by the polynomials in the set or list polys.

The rules laid down in the introduction to the groebner package concerning the polynomial types and the ordering apply.

The polynomials in the list polys must all be of the same type. In particular, do not mix polynomials created via poly and polynomial expressions!

Examples

Example 1

An example from the book of Cox, Little and O'Shea (see below):

groebner::dimension([y^2*z^3, x^5*z^4, x^2*y*z^2]) Parameters

 polys A list or set of polynomials or polynomial expressions of the same type. The coefficients in these polynomials and polynomial expressions can be arbitrary arithmetical expressions. order One of the identifiers DegInvLexOrder, DegreeOrder, and LexOrder, or a user-defined term ordering of type Dom::MonomOrdering. The default ordering is DegInvLexOrder.

Return Values

Nonnegative integer

References

The implemented algorithm is described in Cox, Little, O'Shea: “Ideals, Varieties and Algorithms”, Springer, 1992, Chapter 9.

Algorithms

First, the Gröbner basis of the given polynomials with respect to the given monomial ordering is computed using groebner::gbasis. This Gröbner basis is then used to compute the dimension of the affine variety generated by the polynomials.