Furstenberg--S\'{a}rk\"{o}zy theorem and partition regularity of polynomial equations over finite fields

We prove new combinatorial results about polynomial configurations in large subsets of finite fields. Bergelson--Leibman--McCutcheon (2005) showed that for any polynomial $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, if $A \subseteq \mathbb{F}_q$ is a subset of a $q$-element finite field and $A$ does not contains distinct $a, b$ such that $b - a = P(x)$ for some $x$, then $|A| = o(q)$. In fields of sufficiently large characterstic, the bound $o(q)$ can be improved to $O(q^{1/2})$ by the Weil bound. We match this bound in the low characteristic setting and give a complete algebraic characterization of the class of polynomials for which the Furstenberg--S\'{a}rk\"{o}zy theorem holds over finite fields of fixed characteristic. Our next main result deals with an enhancement of the Furstenberg--S\'{a}rk\"{o}zy theorem over finite fields. Another consequence of the Weil bound is that if $P(x) \in \mathbb{Z}[x]$, $A, B \subseteq \mathbb{F}_q$, and there do not exist elements $a \in A$ and $b \in B$ with $b - a = P(x)$ for some $x$, then $|A| |B| = O(q)$, provided that the characteristic of $\mathbb{F}_q$ is sufficiently large depending on $P$. We provide a complete description of the family of polynomials for which this asymmetric enhancement holds over fields of fixed characteristic, achieving the same quantitative bounds that are available in the high characteristic setting. The exponential sum estimates that we produce in dealing with the above problems also allow us to establish partition regularity of families of polynomial equations over finite fields. As an example, we prove: if $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, then for any $r \in \mathbb{N}$, there exists $N = N(P,r)$ and $c = c(P,r) > 0$ such that if $q > N$ and $\mathbb{F}_q = \bigcup_{i=1}^r{C_i}$, then there are at least $cq^2$ monochromatic solutions to the equation $P(x) + P(y) = P(z)$.