6 problems

Let $P_1,\dots,P_m$ be $m$ polynomials of arbitrary degrees $0\leq d_1,\dots,d_m\leq m-1$ in $(\mathbb{F}[\alpha_1,\dots,\alpha_n])[\alpha]$. Define … where…

Let $B_R\subset{\mathbb R}^n$ be a ball of radius $R$, and let $\Omega$ be a collection of $(R^{1/2},R)$-tubes in $B_R$ with $R^{-1/2}$-separated directions. Suppose the tubes are…

Let $R(n,q)$ denote the maximum cardinality of a subset of the vector space $4\mathbb{F}_q^n4$ containing no right angles. The notation $4\Theta4$ refers to asymptotic equality up…

Let $R$ be a ring, and let $A_1,\dots,A_n\subset R$ be nonempty (but possibly infinite) subsets. Let $\mathcal{H}=\{H_j\}_{j\in J}$ be a covering of the grid … by hyperplanes. The…

Multiplicity joints conjecture. The number of joints counted with multiplicities satisfies

Fix $n$, and let $I^n\subseteq\mathbb{R}^n$ be the unit cube. Given $\varepsilon,H>0$, a positive integer $N$, and a polynomial $Q$, subdivide $I^n$ into cubes of edge length…