5 problems

Let $V$ be a finite-dimensional vector space, let $f\in S^3V$, and let $E\subseteq V^*$ be a linear subspace. Write $f|_E$ for the restriction of $f$ to $E$, and let…

Prime-modulus rank-count conjecture. For all $n$, uniformly in $1\leq H\leq R$, and for every relevant rank threshold $r$,

Davenport's geometric-condition conjecture. If $C$ satisfies Davenport's Geometric Condition, then the asymptotic formula for the number of solutions to $C(\mathbf{x})=0$ holds wit…

HRH. The function $\xi({\mathbf m};s)$ has a meromorphic continuation to $\mathbb C$ of finite order, with only possible poles at $s=\frac{3}{2}$ and $s=\frac{5}{2}$; there is a si…

Let $F\in k[x_0,\ldots,x_n]$ be a generic homogeneous cubic form, and let $cr(F)$ denote its cactus rank and $r(F)$ its rank. Cactus-rank conjecture. The cactus rank equals the ran…