4 problems

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements, and let $P$ be a multilinear polynomial of degree $d$. Write $a_{\mathbb F_q}(P)$ for its analyti…

Let $K$ be a field, let $V=V_1\times\dots\times V_d$, and let $P:V\to K$ be a multilinear polynomial of degree $d$. Write $r_K(P)$ for the Schmidt rank of $P$, and let $\bar K$ be…

Let $k$ be a finite field of characteristic $p$, let $V$ be a finite-dimensional $k$-vector space, and let $P:V\to k$ be a polynomial of degree $d$ and rank greater than $r$. The b…

Let $k$ be a field, let $\bb V$ and $\bb W$ be $k$-vector spaces with $\dim(\bb W)\leq m$, and let $P:\bb V\to k$ be a polynomial of degree $d$. For an affine map…