Projective gradient criterion in characteristic dividing the degree
= Projective gradient criterion in characteristic dividing the degree
For a reduced <projective hypersurface> $X=V(f)$ over an algebraically closed field,
$$
\operatorname{Sing}(X)=X\cap V(f_{x_0},\ldots,f_{x_n}).
$$
The <Euler homogeneous function theorem> makes the intersection redundant only when the characteristic does not divide the degree. In characteristic $p$, the irreducible polynomial $x_0^p+x_1^{p-1}x_2$ has zero gradient at $[1:0:0]$, which is not on $X$.