## `groebner::dimension` -- the dimension of the affine variety generated by polynomials

#### Introduction

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

#### Call(s)

`groebner::dimension(polys <, order>)`

#### Parameters

`polys` | - | a list or set of polynomials of the same type. Alternatively, a list or set of polynomial expressions with rational coefficients.
`order` | - | one of the identifiers DegInvLexOrder, DegreeOrder, and LexOrder, or a user-defined term ordering of type `Dom::MonomOrdering`. The default ordering is DegInvLexOrder.

#### Returns

a nonnegative integer

#### Related Functions

`groebner::gbasis`, `poly`

#### Details

- 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!

#### 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])

                                         2

#### Background

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

#### Changes

- Further term orderings from `Dom::MonomOrdering` are handled. The special term orderings from the Gröbner package were moved there.

Do you have questions or comments?
Copyright © SciFace Software GmbH & Co. KG 2000