Title: Improved estimates for polynomial Roth type theorems in finite fields,
Abstract: We prove that, under certain conditions on the function pair $\varphi_1$ and
$\varphi_2$, bilinear average $p^{-1}\sum_{y\in
\mathbb{F}_p}f_1(x+\varphi_1(y)) f_2(x+\varphi_2(y))$ along curve $(\varphi_1,
\varphi_2)$ satisfies certain decay estimate. As a consequence, Roth type
theorems hold in the setting of finite fields. In particular, if
$\varphi_1,\varphi_2\in \mathbb{F}_p[X]$ with $\varphi_1(0)=\varphi_2(0)=0$ are
linearly independent polynomials, then for any $A\subset \mathbb{F}_p,
|A|=\delta p$ with $\delta>c p^{-\frac{1}{12}}$, there are $\gtrsim
\delta^3p^2$ triplets $x,x+\varphi_1(y), x+\varphi_2(y)\in A$. This extends a
recent result of Bourgain and Chang who initiated this type of problems, and
strengthens the bound in a result of Peluse, who generalized Bourgain and
Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.