#### UserFunction PrimaryDecomposition

PrimaryDecomposition(u:DPOLID(C,v,O))

The single argument $u$ has the type $DPOLID$ which abbreviates
{\em Distributive Polynomial Ideal}, it must represent a zero-dimensional 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.

\vspace*{1mm}\noindent{\bf Specification.}
Returns the primary decomposition of $u$.

\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window.

\5{\tt I:=\{55*x*y*z+36*x+85*y-98*z,41*x*y*z-47*z,25*x*y+7*z\} }

\5{\tt PrimaryDecomposition I;}

\vspace*{2mm}

\vspace*{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{
The given ideal:
\vspace*{3mm}

$<55xyz+36x+85y-98z,41xyz-47z,25xy+7z>$
\vspace*{2mm}

The primary decomposition:
\vspace*{2mm}

$<x+\fracsm{85}{36}y-\fracsm{1433}{1476}z, y^2-\fracsm{1433}{3485}yz-\fracsm{252}{2125}z,z+\fracsm{1175}{287}>\cap$

\vspace*{3mm}\hspace*{30mm}
$<x+\fracsm{85}{36}y-\fracsm{1433}{1476}z, y^2-\fracsm{1433}{3485}yz-\fracsm{252}{2125}z,z>$

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.