Is there any nice way to show an ideal $(x^5,y^6,xy)\subset F[x,y]$ where $F[x,y]$ is a polynomial ring with two variable over a field $F$ cannot be generated by two elements? I.e., there is no $f(x,y),g(x,y)\in F[x,y]$ such that $(f(x,y),g(x,y)) = (x^5,y^6,xy)$.
Edit: Nice way means showing not by case by case argument.
Let $I=(x^5,y^6,xy)$ and $\mathfrak m=(x,y)$.
Suppose that $I$ is generated by two elements, say $f,g$. Then the $F$-vector space $I/\mathfrak mI$ is generated by (the residue classes of) $f,g$ (why?). It follows that $\dim_FI/\mathfrak mI\le2$. In particular, (the residue classes of) $x^5,y^6,xy$ are linearly dependent over $F$, that is, there exist $a,b,c\in F$ not all zero such that $ax^5+by^6+cxy\in\mathfrak mI$. Since $\mathfrak mI$ is a monomial ideal we get that at least one of the monomials $x^5,y^6,xy$ belong to $\mathfrak mI$. It follows that at least one of the monomials $x^5,y^6,xy$ is divisible by one of the monomials which generate $\mathfrak mI$, a contradiction.