Singular

D.4.26.3 primdecGTZE

Usage:

primdecGTZE(I); i ideal

Return:

a list pr of primary ideals and their associated primes for a proper ideal, and an empty list for the unit ideal.

pr[i][1] the i-th primary component, pr[i][2] the i-th prime component.

Note:

- Algorithm of Gianni/Trager/Zacharias.
- Designed for characteristic 0, works also in char k > 0, if it terminates (may result in an infinite loop in small characteristic!) - For local orderings, the result is considered in the localization of the polynomial ring, not in the power series ring
- For local and mixed orderings, the decomposition in the corresponding global ring is returned if the string 'global' is specified as second argument

LIB "primdec.lib";
ring  r = 0,(x,y,z),lp;
poly  p = z2+1;
poly  q = z3+2;
ideal I = p*q^2,y-z2;
list pr = primdecGTZE(I);
pr;
==> [1]:
==>    [1]:
==>       _[1]=z2+1
==>       _[2]=y-z2
==>    [2]:
==>       _[1]=z2+1
==>       _[2]=y-z2
==> [2]:
==>    [1]:
==>       _[1]=z6+4z3+4
==>       _[2]=y-z2
==>    [2]:
==>       _[1]=z3+2
==>       _[2]=y-z2
ideal J = 1;
list prempty = primdecGTZE(J);
prempty;
==> empty list