Critical loci of self-maps of projective space

Let $K$ be an algebraically closed field and let $n,d\ge 2$. We show that the critical scheme of a general endomorphism of ${\mathbb{P}}_K^n$ of degree $d$ is an integral hypersurface. This extends a result of Ingram–Ramadas–Silverman to arbitrary characteristic. We use basic facts about polynomial rings to show that certain carefully chosen examples have absolutely irreducible Jacobian. To handle the wild case where the characteristic of $K$ divides $d$, we introduce a polynomial that imitates the homogeneous Jacobian determinant.