#### UserFunction Saturation

`Saturation(u,v:DPOLID(C,v,O))`

The arguments $u$ and $v$ have the type $DPOLID$ which abbreviates

*Distributive Polynomial Ideal*. The meaning of its parameters is as follows.

$C$: Coefficient type.

$v=\{v_1,v_2,\ldots \}$: Distributive variables in decreasing order.

$O:LEX|GRLEX|GREVLEX$: Term ordering.

**Specification.** The saturation of $u$ w.r.t. $v$ is returned.

**Examples.** The input is given in *Reduce* algebraic mode syntax.

The first example is from Greuel and Pfister, page 83.

`I:={x**5*z**3,x*y*z,y*z**4}; J:={z};`

`T==|DPOLID(Q,{x,y,z},GRLEX)|;`

`Saturation(I,J|T|);`

The output is returned in a separate window.

$<x^5z^3,xyz,yz^4>$

$<z>$

The saturation ideal:

$<y,x^5>$

**Related Functions.** Quotient.

**References.**

W.W. Adams, P. Loustaunau, * An Introduction to Groebner Bases*, American Mathematical Society, 1994.

D. Cox, J. Little, D. O'Shea, * Ideals, Varieties and Algorithms Using Algebraic Geometry*,

Undergraduate Texts in Mathematics, Springer, 1992 and 1998.

G.M. Greuel, G. Pfister, *A Singular Introduction to Commutative Algebra*, Springer, 2002.

M. Kreuzer, L. Robbiano, *Using Computational Commutative Algebra I and II*, Springer, 2000.