# CASA Function: implUnionLCM

Compute the union of algebraic sets.

### Calling Sequence:

### Parameters:

- As : exprseq(algset("impl"))
- algebraic sets in implicit representation

### Result:

- U : algset("impl")
- The union of the given algebraic sets.

### Description:

- The function computes the union of algebraic sets in implicit form by computing the intersection of the corresponding ideals.
- For the case of two algebraic sets A and B: Let the ideal of A be I, a subset of K[x1,...,xn], and let the ideal of B be J. The intersection is computed by the ideal contraction S := intersect(<union(t*I,(t-1)*J)>,K[x1,...,xn]), where t is a new variable and <X> denotes the ideal generated by the set X. A basis for the ideal S can be obtained by a Groebner basis calculation. Furthermore, S = intersect(I,J).
- If the algebraic sets have common components then the multiplicity of these components is not increased in the result. (compare implUnion.)

### Examples:

`> ` **a1 := mkImplAlgSet([x^3+x^2*y-x,z],[x,y,z]);**

`> ` **a2 := mkImplAlgSet([x,y^2+z^2-1],[x,y,z]);**

`> ` **implUnionLCM(a1,a2);**

### See Also:

[CASA]
[implUnion]
[implSubSet]
[implEqual]
[implEmpty]
[implIntersect]
[implIdealQuo]
[equalBaseSpaces]