# Documentation

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# `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.