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# groebner::stronglyIndependentSets

Strongly independent set of variables

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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

[1] Kredel H. and V. Weispfenning, "Computing dimension and independent sets for polynomial ideals", JSC volume 6 (1988), 231-247.