12 problems

Let $\mathbb K$ be a field with characteristic $\chi(\mathbb K)\ne 2$, let $n\geq 4$, and let … be a $3$-form. Its nullity is the dimension of the largest subspace on which the ass…

Let $\mathbb{K}$ be a field, let $d\geq 2$ be an integer, and let … Here $\operatorname{PR}_{\mathbb{K}}(f)$ denotes the partition rank of $f$, and $\overline{\mathbb{K}}$ is an al…

Let the operation act on the finite space $V=K^3$ over a finite field or residue field of size $N$, and consider the resulting orbit decomposition. An orbit has maximal length when…

Hardness conjecture for the discrete iteration problem. Given $a$ and $a^n$, it is computationally hard to recover $n$ for general parameter values, input vectors, and sufficiently…

No-closed-form-expression hypothesis. There is no general closed-form expression for the components of $a^n$ in terms of a fixed polynomial formula with finitely many terms whose s…

Lovett–Adiprasito–Kazhdan–Ziegler conjecture. There exists a constant $C_d$ such that for any finite field and multilinear form $f$ we have

Let $G$ be a vector space over $\mathbb{F}_p$, and let $alpha\colon G^k\tomathbb{F}_p$ be an $r$-approximately symmetric multilinear form, meaning that the partition rank of…

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, let $V_1,\ldots,V_d$ be $\mathbf k$-vector spaces, and let $L\subset (V_1\otimes\cdots\otimes V_d)^\vee$…

Let $\mathbf k$ be a field, let $\bar{\mathbf k}$ be its algebraic closure, and let $P$ be a multilinear $\mathbf k$-polynomial of degree $d\geq 2$. Write $r_{\mathbf k}(P)$ for it…

Let $n$, $k$, $d$, and $t$ be integers such that $n-1\geq d\geq k\geq 2$ and $d\leq t(d-k+1)$. Let … The main theorem establishes that, when $\alpha\leq 3003$, there exists an…

Multidimensional direct sum conjecture. The rank of $d$-dimensional tensors is additive with respect to direct sums:

Let $m,n\geq1$, let ${\bf d}=(d_1,\ldots,d_m)$, and let $\mathcal{M}_{\bf d}$ be the space of complex hypermatrices of type $d_1\times\cdots\times d_m$. For…