Lovett–Adiprasito–Kazhdan–Ziegler partition-rank conjecture

Let $V_1,\ldots,V_d$ be finite-dimensional vector spaces over a finite field $\mathbf{k}$, and let $f:V_1\times\cdots\times V_d\to\mathbf{k}$ be a multilinear form. Define the partition rank by

$$\operatorname{prk}(f)=\min\left\{r:f=\sum_{i=1}^r g_i\cdot h_i\right\},$$

where $g_i$ and $h_i$ are multilinear in disjoint sets of variables. Define the analytic rank by

$$\operatorname{ark}(f)=-\log_q\frac{|Z|}{|V_1\times\cdots\times V_{d-1}|},$$

where

$$Z=\{(x_1,\ldots,x_{d-1})\in V_1\times\cdots\times V_{d-1}:f(x_1,\ldots,x_{d-1},\cdot)\equiv 0\}.$$

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

$$\operatorname{prk}(f)\le C_d\cdot\operatorname{ark}(f).$$

The inequality $\operatorname{ark}(f)\leq\operatorname{prk}(f)$ is known, and the conjecture asserts the reverse inequality up to a constant depending only on $d$.