# CASA Function: implUnion

Computes the union of algebraic sets.

### Calling Sequence:

• U := implUnion(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 product of the corresponding ideals. These ideals are given by a finite basis. A basis of the product is obtained by forming all products of basis elements.
• For the case of two algebraic sets A and B: Let the ideal of A have the basis {p1,...,pn} and let the ideal of B have the basis {q1,...,qm} then the product has the basis {p1*q1,p2*q1,...,pn*q1,...,p1*qm,p2*qm,...,pn*qm}. If the algebraic sets have common components then the result contains these components with higher multiplicity (compare implUnionLCM)

### Examples:

> a1 := mkImplAlgSet([x^3+x^2*y-x,z],[x,y,z]); > a2 := mkImplAlgSet([x,y^2+z^2-1],[x,y,z]); > implUnion(a1,a2);  