14 problems

Let $F\in K[x_0,\ldots,x_n]$ be a general cubic form, and let its cactus rank be the minimal length of a finite apolar subscheme, allowing singularities. In fact, its cactus rank i…

Let $\mathcal{H}_d$ denote the class of stable homogeneous polynomials of degree $d$, and let $\mathcal{H}_d/\{y^2\}$ denote the corresponding initial segment modulo $y^2$. The sta…

Let $K\subset \mathbb R^{d}$ be a convex body centrally symmetric about the origin, and let $\operatorname{Bd}(K)$ denote its boundary. For a continuous function…

Two-variable Chowla conjecture. The sequence $\lambda(f(m,n))$ has average value $0$ as $(m,n)$ ranges over lattice points in any sector of the plane with vertex at the origin. Thi…

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 $f\in\mathcal{R}_{n,d}$ be a homogeneous polynomial, and let its Waring locus be the set of linear forms, viewed as points of $\mathbb{P}((\mathbb{C}^n)^*)$, that occur in a mi…

Conjecture on the $P-1$ Fischer pair. The operator $F_P$ is a bijection on $E(\mathbb{C}^{d})$; equivalently, $P-1$ and $P^{\ast}(D)$ form a Fischer pair for $E(\mathbb{C}^{d})$. T…

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…

Let $k$ be a field of characteristic $p$, and let $d\geq 4$ be an integer not divisible by $p$. Let $e$ be the smallest positive integer such that … For a rational number, write…

Let $f\in R=k[z_0,\dots,z_n]$ be a homogeneous polynomial. Let $\operatorname{MCM\mathchar`-rank}(f)$ be the minimum of the integers $r$ for which there is a reduced, homogeneous m…

Distance-to-discriminant conjecture. One has

Regularly spaced roots conjecture. The polynomial of degree $d$ with $2d$ regularly spaced roots on the unit circle is a global maximum of the distance to the discriminant among po…

Carlson–Griffiths conjecture. If $V(f)$ does not possess singularities which are large in relation to the degree, then $f$ is determined by $E(f)$, up to a nonzero scalar multiple.

Morozov–Shakirov conjecture. The integral of the exponential of $P$ is always a generalized hypergeometric function of algebraic invariants of the form.