Let $F\in\mathbb{C}[X,Y]$ be an irreducible polynomial and $n\in \mathbb{N}$, $n\ge1$, $p_i\in\mathbb{C}[X]$ for $0\le i\le n$, such that $F(X,Y)=\sum\limits_{i=0}^{n}p_i(X)Y^{n-i}.$ Let $x\in\mathbb{C}$, such that $p_0(x)\ne0$, so $F(x,Y)$ has $n$ zeros, and one of these $n$ zeros is a single one (so there is a $y\in\mathbb{C}$, such that $F(x,y)=0$ and $\frac{\partial F}{\partial Y}(x,y)\ne0$). Is it possible that the polynomial $F(x,Y)$ has a multiple zero?
Thanks in advance.