Tridimensional character sums with polynomial arguments and applications

Let $p$ be a large prime and $\chi$ a non-trivial Dirichlet character modulo $p$. We study the character sum $$\sum _{a\sim A}\sum _{b\sim B}\sum _{c\sim C}\alpha (a,b)\beta (c)\chi (f(a)+bc),$$ where $z\sim Z$ means $Z\le z<2Z$, $f\in \mathbb{Z}[X]$ is of small degree $k$, and $\mathit{\alpha }=(\alpha (a,b){)}_{a\sim A,b\sim B}$ and $\mathit{\beta }=(\beta (c){)}_{c\sim C}$ are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) $\mathit{\beta }\equiv 1$, $k=2,3,4,5$ and $A,B,C>p^{\frac{1}{8}+\epsilon }$, (2) $\mathit{\beta }$ general, $k=2,3$ and $A,B,C>p^{\frac{1}{6}+\epsilon }$, where $\epsilon >0$ is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting $2\times 2$ matrices over ${\mathbb{F}}_p$ with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.