3 problems

Let $K$ be a field of characteristic $0$, let $\varphi\in K[x_1,\ldots,x_n]$ be a polynomial whose Hessian determinant is a nonzero constant, and let $\bar\varphi$ be its formal Le…

Let $\Bbbk$ be a field of characteristic zero and let $\varphi\in\Bbbk[x_1,\dots,x_n]$. Write … If $\det\operatorname{Hess}\varphi\in\Bbbk^{\times}$, the gradient map has a formal…

Dolgachev's conjecture. A homaloidal polynomial whose multiplicative Legendre transform is a polynomial must be a relative invariant of a regular prehomogeneous vector space.