This talk is a survey on techniques for estimating the number of solutions to polynomial equations over a finite field $\mathbb{F}_q$. We use exponential sums-constructed from additive and multiplicative character-to establish these counts. By analyzing the algebraic and orthogonality properties of Gaussian and Jacobi sums, we build a mathematical framework to derive bounds for equations over $\mathbb{F}_q$ and its extensions.
A central part of the survey is the study of Dirichlet $L$-functions over the polynomial ring $\mathbb{F}_q[X]$. We demonstrate how these $L$-functions can be used with characters on the set of monic rational functions to get bounds on the absolute value of character sums on polynomials. We discuss the Weil bounds for absolutely irreducible bivariate polynomials, focusing on two primary special cases:
These elementary polynomial methods are emphasized because they achieve the same results as advanced algebraic geometry without requiring its full abstract machinery.
The final part of the talk details an algorithmic application of these bounds: decoding codes constructed on evaluating polynomial composed with multiplicative character on whole field and dual BCH codes from a constant fraction of errors. This problem is challenging because character evaluations (like the quadratic character $\eta$) discard most information about a polynomial, leading to underdetermined systems where traditional decoding fails. To address this, we introduce pseudopolynomials which are of high-degree polynomials whose Hasse derivatives agree with low-degree polynomials when evaluated on $\mathbb{F}_q$. By merging the Berlekamp–Welch like arguments for Reed-Solomon with the interpolation techniques from Stepanov's method, to decode from $\frac18$th of the minimum distance of the codes. The algorithm is from a recent work of Kopparty.
