## Divisibility of Exponential Sums

### Publications

*The Covering Method for Exponential Sums and Some Applications*. (2020). Ivelisse M. Rubio.

**Abstract:**Exponential sums over finite fields are an important tool for solving mathematical problems and have applications to many other areas. However, some of the methods and proofs of the results are nonelementary. The main purpose of this article is to present the covering method, an elementary and intuitive way to estimate or compute the $p$-divisibility of exponential sums, which is particularly convenient in the applications. The covering method allows us to determine solvability of systems of polynomial equations, improve the search for balanced Boolean functions, give better estimates for covering radius of codes, and has many other applications.Final version published in the Notices of the American Mathematical Society, May 2020.

*Exact $2$-divisibility of exponential sums associated to boolean functions*. (2018). Francis N. Castro, Luis A. Medina, & Ivelisse M. Rubio.

**Abstract:**In this paper we extend the covering method for computing the exact $2$-divisibility of exponential sums of Boolean functions, improve results on the divisibility of the Hamming weight of deformations of Boolean functions, and provide criteria to obtain non-balanced functions. In particular, we present criteria to determine cosets of Reed-Muller codes that do not contain any balanced function, and to construct deformations of symmetric functions that are not balanced. The use of the covering method together with classifications of cosets of Reed-Muller codes obtained by the action of linear groups can improve the search of balanced functions in Reed-Muller codes dramaticallyFinal version published in Cryptography and Communications, 10(4), 655-666, 2018

*Exact p-Divisibility of Exponential Sums via the Covering Method*. (2015). Francis N. Castro, & Ivelisse M. Rubio.

**Abstract:**In general, the methods to estimate the $p$-divisibility of exponen- tial sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact p-divisibility of exponential sums over a prime field. The results here allow us to compute the exact $p$-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.Final version published in Proceedings of the American Mathematical Society 143 (2015) 1043-1056

*Exact Divisibility of Exponential Sums over the Binary Field via the Covering Method*. (2011). Francis N. Castro, Luis A. Medina, & Ivelisse M. Rubio.

**Abstract**: Boolean functions are one of the most studied objects in mathematics. In this paper, we use the covering method to compute the exact 2-divisibility of exponential sums of boolean functions with prescribed leading monomials. Our results generalize those of [6] and [5] for the binary field. As an application of our findings, we provide families of boolean functions that are not balanced, and give sufficient conditions for the solvability of systems of boolean equations.

Final version published in AMS Contemporary Math, Volume 537 (2011), pp. 129-136.

*Divisibility of Exponential Sums via Elementary Methods*. (2010). Francis N. Castro, Hugues Randriam, Ivelisse Rubio & H. F. Mattson, Jr.

**Abstract**: We present a totally elementary method for evaluating the order of p-divisibility of exponential sums over a prime field. This method unifies and sometimes improves many previously known results, such as those of Ax-Katz, Moreno-Moreno, Adolphson-Sperber, and Cao-Sun.

Final version published in Journal of Number Theory 130 (2010), pp. 1520-1536.

*Divisibility of Exponential Sums and Solvability of Certain Equations Over Finite Fields*. (2007). Francis N. Castro, Ivelisse M. Rubio, & José M. Vega.

**Abstract**: In [3], Carlitz determined conditions under which in finite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz’s results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.

Final version published in Quart. J. Math. 00 (2007), 1–13; doi:10.1093/qmath/han013.

### Student Projects

#### Undergraduate Students

*Exact divisibility of exponential sums associated to elementary symmetric Boolean functions*. (2017). Oscar E. González, Raúl E. Negrón, Francis N. Castro, Luis A. Medina & Ivelisse M. Rubio.

[Poster]

**Abstract:**In this paper, we present an elementary method to compute or estimate the exact 2-divisibility of exponential sums associated to symmetric Boolean functions. As a direct consequence of these results, we prove some of the open cases of Cusick-Li-Stănică’s conjecture about balanced symmetric Boolean functions.

*Exact $p$-divisibility of exponential sums of polynomials over finite fields*. Poster. (2015). Ramón L. Collazo, Julio J. del Cruz, Daniel E. Ramírez, Francis N. Castro & Ivelisse M. Rubio.

**Abstract:**An $n$-variable Boolean function F is a function defined over $F^n$ with values in $F$, the finite field with two elements. Our aim is to calculate the exact $2$ divisibility of exponential sums associated to Boolean functions. This allows us to determine whether a Boolean function is balanced, i.e. whether $|\{x \in F|f(x) = 1\}| = 2^{n−1}$, which can be difficult in general. We prove two theorems which give affirmative answers to some of the open cases of Cusick-Li-Stănică’s conjecture about the non-existence of certain balanced Boolean funtions.

## Solvability of Equations over Finite Fields

### Publications

*An improvement of a theorem of Carlitz*. (2020). Francis N. Castro, Oscar Moreno & Ivelisse M. Rubio.

**Abstract**: We improve a result of Carlitz about the number of variables needed for a system of polynomial equations with coefficients in $F_q [X]$ to have non-trivial solutions by considering the $p$-weight degree of the polynomials. By providing infinite families of polynomials we illustrate that our improvement is significant and, in general, is tight.

Final version published in Journal of Pure and Applied Algebra 224 (2020), https://doi.org/10.1016/j.jpaa.2019.106246.

*Construction of systems of polynomial equations with exact $p$-Divisibility via the covering method*. (2014). Francis N. Castro, & Ivelisse M. Rubio.

**Abstract**: We present an elementary method to compute the exact $p$-divisibility of exponential

sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

Final version published in JJournal of Algebra and its Applications, Vol. 13, No. 6 (2014) 1450013 (15 pages). This paper was selected as one of the best papers published in the JAA journal in 2014.

*Diagonal Equations, Section in the Handbook of Finite Fields*(2013). Francis Castro, & Ivelisse Rubio. Editors: Gary Mullen and Daniel Panario; CRC Press

*Solvability of Systems of Polynomial Equations with Some Prescribed Monomials*. (2010). Ivelisse M. Rubio & Francis N. Castro.

**Abstract**: We prove that, under some natural conditions, given a system of polynomials F_{1}, … , F_{t}with monomials of disjoint support, any system F_{1}+ G_{1}, … , F_{t}+ G_{t}, where the p-weight degree of the G_{i}‘s is smaller than the degree of the monomials in the F_{i}‘s, is solvable. This generalizes a result of Carlitz. As byproduct we also compute the exact p-divisibility of the number of solutions of the system.

Final version published in Contemporary Mathematics, vol. 518 (2010), Amer. Math. Soc., Providence, RI, pp. 73-81.

*On Systems of Linear and Diagonal Equation of Degree p*. (2008). Francis N. Castro, Ivelisse M. Rubio, Puhua Guan, & Raúl Figueroa.^{i}+ 1 Over Finite Fields of Characteristic p

**Abstract**: One of the most important questions in number theory is to find properties on a system of equations that guarantee solutions over a field. A well-known problem is Waring’s problem that is to find the minimum number of variables such that the equation x_{1}^{d}+ ··· + x_{n}^{d}= β has solution for any natural number β. In this note we consider a generalization of Waring’s problem over finite fields: To find the minimum number δ(k, d, p^{f}) of variables such that a system*x*_{1}^{k}+ ··· + x_{n}^{k}= β_{1},

x_{1}^{d}+ ··· + x_{n}^{d}= β_{2}*has solution over***F**_{p f}for any (β_{1},β_{2}) ∈**F**^{2}_{p f}. We prove that, for p > 3,

δ(1, p^{i}+1, p^{f}) = 3 if and only if f ≠ 2i. We also give an example that proves that, for p = 3, δ(1, 3^{i}+1, 3^{f}) ≥ 4.

Final version published in Finite Fields Appl., 14 (2008), 648-657.

*Divisibility of Exponential Sums and Solvability of Certain Equations Over Finite Fields*. (2007). Francis N. Castro, Ivelisse M. Rubio, & José M. Vega.

**Abstract**: In [3], Carlitz determined conditions under which in finite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz’s results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.

Final version published in Quart. J. Math. 00 (2007), 1–13; doi:10.1093/qmath/han013.

### Student Projects

#### Master Students

*Solvability of systems of polynomial equations with multivariate polynomials as coefficients*. Thesis. (2020). Carlos Seda.

**Abstract:**In [3] Castro, Moreno and Rubio generalize the results of Moreno-Moreno’s theorem that gives a bound for the power of a prime p to divide the number of common zeros of the multivariate polynomials $F_1, …, F_t$ over a finite field $F_q$. This generalization regarded the coefficients of the polynomials to be uni-variate polynomials over a finite field instead of plain elements of the finite field. The result led to improve a theorem of Carlitz, for the estimation of the number of variables needed so that a system of polynomial equations with coefficients in $F_q[X]$ can have non-trivial zeros. We generalize the results of Castro, Moreno and Rubio to polynomials whose coefficients are multivariate polynomials over finite fields.

#### Undergraduate Students

*Solvability of Systems of Polynomial Equations over Finite Fields*. Technical Report. (2014). Ramón L. Collazo, Julio J. de la Cruz & Daniel E. Ramírez.

**Abstract:**An important problem in mathematics is to determine if a system of polynomial equations has or not solutions over a given set. We study systems of polynomial equations over finite fields $F_p$, $p$ prime, and look for sufficient conditions that guarantee their solvability over the field. Using the covering method of (Castro & Rubio, n.d.) we get conditions on the degrees of the terms that allow us to construct families of systems that have exact p-divisibility of the number of solutions and therefore guarantee the solvability of the system over the finite field.

*Número de Waring en Cuerpos Finitos*. Technical Report. (2011). Zahir Mejias & Jean-Karlo Accetta.

[Poster] [Slideshow]

**Abstract:**El número de Waring es el número mínimo de variables que necesita una ecuación de la forma x_{1}^{d}+ ··· + x_{n}^{d}= β para que tenga solución en los números naturales para cualquier valor de la constante en los números naturales. Trabajamos en el estudio de generalizaciones de este problema cuando los valores de la constante y las soluciones se definen en cuerpos finitos. Hemos desarrollado un programa para calcular el número de Waring, y con el mismo hemos mejorado en resultados anteriormente publicados.