14 problems

Sharp upper-bound conjecture. One should have

For a 3-graph $G$ on $[n]$, define … Let $DI(n,\ell)$ be the ideal of polynomials whose relevant partial derivatives vanish whenever any $\ell$ variables are identified, and let…

Sauermann and Wigderson's conjecture. The upper bound is tight when $n$ is sufficiently large with respect to $k$; that is, for sufficiently large $n$ relative to $k$,…

Let $S elangle0,1|^ranglen$ be a symmetric subset of the Boolean hypercube, and let $Wn(S)$ be its set of Hamming weights. Let $W{n,2}$ denote the corresponding weight set for the…

Let $p:\{-1,1\}^n\to\mathbb{R}$ be a polynomial of degree at most $d$, with supremum norm $\|p\|_\infty\leq 1$. A variable $i$ has influence $\operatorname{Inf}_i(p)$, and…

Linear-factor witness conjecture. For $d\in[0,n]$ and $E\subseteq[0,n]$, if $j\in[0,n]$ and $j\notin\text{Z-cl}_{n,d}(E)$, then there exists a polynomial…

Generalized-monomial witness conjecture. For $d\in[0,n]$ and $E\subseteq[0,n]$, if $j\in[0,n]$ and $j\notin\text{Z-cl}_{n,d}(E)$, then there exists a polynomial…

The conjecture on removing the modulo condition. The preceding statement holds without the modulo $\Psi[X]$; that is, the equality above holds as an equality of forms rather than o…

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…