Session S26 - Finite fields and applications

Friday, July 16, 12:30 ~ 12:50 UTC-3

Estimates on the number of rational solutions of variants of diagonal equations over finite fields

Mariana Valeria Pérez

Universidad Nacional de Hurlingham-CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Several problems of coding theory, cryptography and combinatorics require the study of the set of $\mathbb{F}_{\hskip-0.7mm q}$-rational points (i.e. points with coordinates in the finite field of $q$ elements $\mathbb{F}_{\hskip-0.7mm q}$) of varieties defined over $\mathbb{F}_{\hskip-0.7mm q}$ on which the symmetric group of permutations of the coordinates acts (see, for example, [1], [2] and [3]).

In this work, we study the set of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of equations defined by polynomials evaluated in power-sum polynomials with coefficients in $\mathbb{F}_{\hskip-0.7mm q}$. More precisely, we consider $m$th-power sum polynomials in the variables $X_1,\ldots,X_n$, namely, the polynomials of the form $P_m=X_1^m+\cdots+X_n^m.$ Let $f \in \mathbb{F}_{\hskip-0.7mm q}[Y_1,\ldots,Y_d]$ and $P_{m_1},\cdots,P_{m_d}\in \mathbb{F}_{\hskip-0.7mm q}[X_1,\ldots,X_n]$. We define the $\mathbb{F}_{\hskip-0.7mm q}$-affine hypersurface given by $f(P_{m_1}, \dots, P_{m_d}) + g$, where $g\in\mathbb{F}_{\hskip-0.7mm q}[X_1,\ldots,X_n]$. Under certain hypotheses on $f$ and $g$, we prove that this hypersurface is absolutely irreducible, and we obtain an upper bound of the dimension of its singular locus. These results are used to obtain estimates on the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational points of this type of hypersurface by applying estimates for absolutely irreducible singular projective varieties provided in [4].

Finally we apply this methodology to the problem of estimating the number of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions of certain polynomial equations on $\mathbb{F}_{\hskip-0.7mm q}$. More precisely, we provide improved estimates and existence results of $\mathbb{F}_{\hskip-0.7mm q}$-rational solutions to the following equations: deformed diagonal equations, generalized Markoff-Hurwitz-type equations and Carlitz's equations (see, for example, [5]).

REFERENCES

[1] A. Cafure, G. Matera and M. Privitelli. Singularities of symmetric hypersurfaces and Reed-Solomon codes, Adv. Math. Commun. 6 (2012). no. 1, 69--94.

[2] E.Cesaratto, G. Matera , M. Pérez and Melina Privitelli. On the value set of small families of polynomials over a finite field. I. J. Combin. Theory Ser. A 124 (2014), 203--227.

[3] G. Matera, M.Pérez and M. Privitelli. Factorization patterns on nonlinear families of univariate polynomials over a finite field, J Algebr. Comb. (2019), 1-51.

[4] S. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J. 2 (2002), no. 3, 589--631.

[5] G. Mullen y D. Panario, Handbook of finite fields. CRC Press, Boca Raton, FL, 2013.

Joint work with Melina Privitelli (Universidad Nacional de General Sarmiento-CONICET, Argentina).

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