Polynomial patterns in subsets of large finite fields of low characteristic

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemer\'{e}di theorem for certain families of polynomials. Namely, we show that if $P_1, \dots, P_m \in (\mathbb{F}_p[t])[y]$ satisfy an equidistribution condition, which is a natural variant of the independence condition in (Peluse, 2019) for our context, then there exists $\gamma > 0$ such that for any $q = p^k$ and any $A_0, A_1, \dots, A_m \subseteq \mathbb{F}_q$, \begin{align*} \left| \left\{ (x,y) \in \mathbb{F}_q^2 : x \in A_0, x + P_1(y) \in A_1, \dots, x + P_m(y) \in A_m \right\} \right| = q^{-(m-1)} \prod_{i=0}^m{|A_i|} + O_{q \to \infty; P_1, \dots, P_m} \left( |A_0|^{1/2} q^{3/2 - \gamma} \right). \end{align*} In particular, if $A \subseteq \mathbb{F}_q$ contains no pattern $\{x, x + P_1(y), \dots, x + P_m(y)\}$ of cardinality $m+1$, then \begin{align*} |A| \ll_{P_1, \dots, P_m} q^{1 - \gamma/ \left( m + \frac{1}{2} \right)}. \end{align*}