# `groebner`::`stronglyIndependentSets`

Strongly independent set of variables

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## Syntax

```groebner::stronglyIndependentSets(`G`)
```

## Description

`groebner::stronglyIndependentSets(G)` computes a strongly independent set of variables modulo the ideal generated by `G`.

A set of variables S is strongly independent modulo an ideal I if no leading term of an element of the Gröbner basis of I consists entirely of elements of S. A set is maximally strongly independent if no proper superset of it is strongly independent. Two maximally strongly independent set may be of different size.

`groebner::stronglyIndependentSets` accepts Gröbner bases in the format returned by `groebner::gbasis`.

## Examples

### Example 1

The following example has been given by Moeller and Mora in 1983.

```G:=map([X0^8*X2, X0*X3, X1^8*X3, X1^7*X3^2, X1^6*X3^3, X1^5*X3^4, X1^4*X3^5, X1^3*X3^6, X1^2*X3^7, X1*X3^8], poly, [X3, X2, X1, X0]): groebner::stronglyIndependentSets(G)```
` `
`delete G:`

## Parameters

 `G` The Gröbner basis of an ideal: a list.

## Return Values

List of the form `[d, S, M]`, where `d` is an integer equal to the dimension of the ideal generated by `G`, `S` is the greatest strongly independent set of variables, and `M` is a set consisting of all maximal strongly independent sets of variables or a piecewise consisting of such lists.

## References

 Kredel H. and V. Weispfenning, “Computing dimension and independent sets for polynomial ideals”, JSC volume 6 (1988), 231-247.