# CASA Function: implIdealQuo

Computes the Zariski closure of the difference of two algebraic sets.

### Calling Sequence:

• C := implIdealQuo(A,B)

### Parameters:

A : algset("impl")
• an algebraic set in implicit representation
B : algset("impl")
• an algebraic set in implicit representation

### Result:

C : algset("impl")
• the Zariski closure of the difference of the two given algebraic sets.

### Description:

• The function computes the algebraic set of the ideal quotient of the ideals of the given algebraic sets.
• The computation is reduced to the computation of the intersection of ideals.
• Let {q1,..,qn} be a basis for I(B), the ideal of B. It holds that I(A):I(B) = Intersection of I(A):<qi>, i=1,...,n, where <qi> is the principle ideal generated by qi. Each I(A):<qi> can be computed again by ideal intersection. Let {p1,...,pm} be a basis for intersect(I(A),<qi>) then {p1/qi,...,pm/qi} is a basis for I(A):<qi>.

### Examples:

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

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

> implIdealQuo(a1,a2);

> implIdealQuo(a1,a1);