4 problems

Let ${\bf x}^m=\{x_1,\ldots,x_m\}$ and let ${\bf 1}^m$ denote the $m$-tuple all of whose entries are $1$. Let $U_n({\bf x}^m,{\bf y}^m)$ be the function defined from the symmetric…

Let $S=\mathbb C[x_1,\ldots,x_n]$ and let $g\in S$ be homogeneous of degree $d\geq 3$, with $n\geq 2$. Assume that its Hessian $H(g)$ is a monomial. Smooth monomial-Hessian conject…

Ciliberto–Ottaviani conjecture. The map $h_{d,n}$ induces a birational morphism onto its image when $d\geq 3$, $n\geq 2$, and $(d,n)\neq(3,2)$. This conjecture concerns the biratio…

Distance-to-discriminant conjecture. One has