9 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…

A positive natural number is sufficiently large if it exceeds some fixed threshold. Four positive cubes conjecture. Every sufficiently large positive natural number can be written…

Geometric factorization conjecture. This geometric picture should hold essentially for any $L(q,P)$ model, thereby explaining the occurrence of $r>0$ in non-complete models.

Variable-count conjecture. One should have sufficient variables whenever

Let $w,x,y,z\in\mathbb N$. Floored cubic-sum conjecture. Every natural number belongs to each of the sets … and … This proposes universal representations by four nonnegative cubic…

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…

A homogeneous cubic equation is a homogeneous polynomial equation of degree $3$ in a specified number of variables, and it is soluble in integers if it has an integer solution othe…