$K[x,y]$ (where $K$ is a field) have any bound for the number of generators of ideals?

Question

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.
But is there any bound for the number of generators of arbitrary ideals? (For example in $K[x,y]$.)

Answer 1

No, $(x,y)^n = (x^n,x^{n-1} y,\dotsc,x y^{n-1},y^n)$ is an ideal of $K[x,y]$ which can not be generated by $n$ elements.