Proposition

Let $f$ and $g$ be distinct non-constant polynomials in $k[x, y]$ and let $I = \ev{f^2, g^3}$. Is it necessarily true that $\sqrt{I} = \ev{f, g}$?

Solution

No. Let $f = x^3, g = x^4$. Then $I = \ev{x^6, x^{12}}$. Clearly, $x \in \sqrt{I}$.

For every $h_1, h_2 \in k[x, y]$, $h_1x^6 + h_2x^{12} = x^6(h_1 + h_2x^6)$, so $x \notin \ev{f, g}$.