# CASA Function: implUnionLCM

Compute the union of algebraic sets.

### Calling Sequence:

• U := implUnionLCM(As)

### 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);