When do partial derivatives $p\_x$, $p\_y$ of a polynomial over $\mathbb{C}$ not have any common factor? Is there a general approach for any number of variables, aka when is the variety defined by the partial derivatives vanishing (the would-be singular points) a finite point set? I.e. when in $\mathbb{C}[x, y, \dots]$ is changing the constant term not enough to render a variety nonsingular? Which means, when does $\mathbb{C}[x, y, \dots]^{\text{sing}}$, polynomials defining singular varieties, which we know is a Zariski closed subspace of generic codimension $\ge$ the number of variables $-\,2$, contain an affine hyperplane in $\mathbb{C}[x, y, \dots]$ which allows the constant to vary?



When do partial derivatives $p_x$, $p_y$ of a polynomial over $\mathbb{C}$ not have any common factor?
In the case of two variables, my example is the only one. Let $g$ be an irreducible polynomial that divides both $p\_x,p\_y$. On the curve $g=0$, we have $dp = p\_xdx+p\_ydy=0$ so $p$ is constant, say $p=c$ on this is curve, so $g$ divides $p-c$. Let's write $p-c = gh$ for some $h$. Then, again on $g=0$, $0=p\_x = g\_xh, 0=p\_y=g\_yh$. But $g\_x,g\_y$ cannot both vanish on $g=0$, so $h$ is divisible by $g$ and $p-c$ is divisible by $g^2$.


