On the number of solutions of systems of certain diagonal equations over finite fields

MELINA PRIVITELLI; MARIANA PÉREZ

Congreso; Mathematical Congress of Americas; 2021

Facultad de Ciencias Exactas y Naturales, UBA

Let Fq be the finite field of q elements. It is a classical problem to determine or to estimate the number N of Fq--rational solutions (i.e. solutions with coordinates in Fq) of systems of polynomial equations over Fq (see, e.g., [5]). There are explicit formulas for the number N only for some very particular cases (see, e.g., [1] and [8]) . For this reason, it is important to have estimates on the number N and results which guarantee the existence of this kind of solutions. In our work, we study systems of diagonal equations. This kind of systems have been considered in the literature because the study of its set of Fq--rational solutions has several applications to different areas of mathematics, such as the theory of cyclotomy, Waring's problem and the graph coloring problem (see, e.g. [3] and [5]). Additionally, information on the number N is very useful in different aspects of coding theory such as the weight distribution of some cyclic codes ([9] and [10]) and the covering radius of certain cyclic codes ( [2] and [4]).In comparison with a single diagonal equation, there are much fewer results about the number of Fq-rational solutions of systems of diagonal equations and most of them use tools involving character sums. In this work, we approach this problem using tools of algebraic geometry. More precisely, we consider an Fq-variety V associated to the system. We study the geometric properties of V, where the key point is obtaining upper bounds of the dimension of its singular locus. This study allows us to obtain estimates and existence results of rational solutions of systems of diagonal equations which in particular improve W. Spackman´s (see [6] and [7]). Furthermore, our techniques can be applied to the study of some variants of these systems such as systems of Dicksons´ equations and generalized diagonal equations.References.[1] X. Cao, W-S. Chou and J. Gu, On the number of solutions of certain diagonal equations over finite fields, Finite Fields Appl. 42 (2016), 225--252.[2] T. Helleseth, On the covering radius of cyclic linear codes and arithmetic codes, Discrete Appl. Math. 11(1985), no. 2, 157--173.[3] R. Lidl and H. Niederreiter, Finite fields, Addison--Wesley, Reading, Massachusetts, 1983.[4] O. Moreno and F. N. Castro, Divisibility properties for covering radius of certain cyclic codes, IEEE Trans. Inform. Theory 49 (2003), no. 12, 3299--3303.[5] Gary L. Mullen and Daniel Panario, Handbook of Finite Fields (1st ed.), Chapman and Hall/CRC, 2013.[6] K. W. Spackman, Simultaneous solutions to diagonal equations over finite fields, J. Number Theory 11(1979), no. 1, 100--115.[7] K. W. Spackman, On the number and distribution of simultaneous solutions to diagonal congruences, Canadian J. Math. 33 (1981), no. 2, 421--436.[8] J. Wolfmann, Some systems of diagonal equations over finite fields, Finite Fields Appl. 4(1998), no. 1, 29--37.[9] X. Zeng L. Hu, W. Jiang, Q. Yue and X. Cao, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl. 16 (2010), no.1, 56--73.[10] D. Zheng, X. Wang, X. Zeng and L. Hu, The weight distribution of a family of p-ary cyclic codes, Des. Codes Cryptogr. 75(2015), no. 2, 263--275.