Francis N. Castro

Tight Bounds For Chevalley–Warning–Ax–Katz Type Estimates, with Improved Applications

Following the work of Chevalley and Warning, Ax obtained a bound on the

Divisibility properties for covering radius of certain cyclic codes

p

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ's conjecture

-divisibility of exponential sums involving multi-variable polynomials of fixed degree

Modular periodicity of exponential sums of symmetric Boolean functions

d

On the covering radius of certain cyclic codes

over a finite field of characteristic

Improvement to Moreno–Morenoʼs theorems

p

Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields

. This bound was subsequently improved by Katz. More recently, Moreno and Moreno, and Adolphson and Sperber, derived bounds that in many instances improved upon the Ax?Katz result. Here we derive a tight bound on the

New families of balanced symmetric functions and a generalization of Cusick, Li and Stǎnicǎ’s conjecture

p

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

-divisibility of the exponential sums. While exact computation of this bound requires the solution of a system of modular equations, approximations are provided which in several classes of examples, improve on the results of Chevalley and Warning, Ax and Katz, Adolphson and Sperber, and Moreno and Moreno. All of the above results readily translate into bounds on the

The p-adic Valuation of Eulerian Numbers: Trees and Bernoulli Numbers

p