Is this true: Every ideal of $K[x_1,\ldots,x_n]$ is generated by some subset with $\leq n$ elements?
It is true when $n=1$, since $K[x]$ is a PID.
I'm trying to prove it is not true for $n\geq2$, via the example $I:=\langle x^2,xy,y^2\rangle\unlhd K[x,y]$.
Does the SINGULAR code below confirm that $I$ is not generated by $1$ or $2$ polynomials?
ring R=0,(x,y),ls; ideal I=x2,xy,y2; minbase(I);
From the SINGULAR manual:
Syntax: minbase ( ideal_expression ) minbase ( module_expression )
Type: the same as the type of the argument
Purpose: returns a minimal set of generators of an ideal, resp. module, if the input is either homogeneous or if the ordering is local.
What does a "minimal set of generators" mean? Minimal w.r.t. cardinality, or w.r.t. $\subseteq$?