3.3 Finite solvability

Lemma 3..2   The set of polynomials $F \subseteq K[x_1,...,x_n]$ has only finitly many solutions iff $\forall 1 \leq i \leq n:\exists f \in std(F):L(f)$ is a power of x_i. (See remark 2.11.)

SINGULAR example:

// consider a line and a plane:
ring R=0,(x,y,z),dp;
number a,b,c,d,e=1,1,1,1,1;
ideal L=a*x+b*y,z;
poly P=c*x+d*y+e*z;
ideal I=L,P;
std(I); // the zero set is a line
a=0;
L=a*x+b*y,z;
I=L,P;
std(I); // the zero set is finite (a point)