Singular

D.8.6.5 zeroSet

Usage:

zeroSet(I [,opt] ); I=ideal, opt=integer

Purpose:

compute the zero-set of the zero-dim. ideal I, in a finite extension of the ground field.

Return:

ring, a polynomial ring over an extension field of the ground field, containing a list 'theZeroset', a polynomial 'newA', and an ideal 'id':

- 'theZeroset' is the list of the zeros of the ideal I, each zero is an ideal. - if the ground field is Q(b) and the extension field is Q(a), then 'newA' is the representation of b in Q(a). If the basering contains a parameter 'a' and the minpoly remains unchanged then 'newA' = 'a'. If the basering does not contain a parameter then 'newA' = 'a' (default). - 'id' is the ideal I in Q(a)[x_1,...] (a' substituted by 'newA')

Assume:

dim(I) = 0, and ground field to be Q or a simple extension of Q given by a minpoly.

Options:

opt = 0: no primary decomposition (default)
opt > 0: primary decomposition

Note:

If I contains an algebraic number (parameter) then I must be transformed w.r.t. 'newA' in the new ring.

LIB "zeroset.lib";
ring R = (0,a), (x,y,z), lp;
minpoly = a2 + 1;
ideal I = x2 - 1/2, a*z - 1, y - 2;
def T = zeroSet(I);
setring T;
minpoly;
==> (4a4+4a2+9)
newA;
==> (1/3a3+5/6a)
id;
==> id[1]=(1/3a3+5/6a)*z-1
==> id[2]=y-2
==> id[3]=2*x2-1
theZeroset;
==> [1]:
==>    _[1]=(1/3a3-1/6a)
==>    _[2]=2
==>    _[3]=(-1/3a3-5/6a)
==> [2]:
==>    _[1]=(-1/3a3+1/6a)
==>    _[2]=2
==>    _[3]=(-1/3a3-5/6a)
map F1 = basering, theZeroset[1];
map F2 = basering, theZeroset[2];
F1(id);
==> _[1]=0
==> _[2]=0
==> _[3]=0
F2(id);
==> _[1]=0
==> _[2]=0
==> _[3]=0