Singularities of Symmetric Hypersurfaces and Reed-Solomon Codes

We determine conditions on q for the nonexistence of deep holes of the standard Reed-Solomon code of dimension k over Fq generated by polynomials of degree k + d. Our conditions rely on the existence of q-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over Fq. We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of q-rational points is established.Fil: Cafure, Antonio Artemio. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Privitelli, Melina Lorena. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Smooth symmetric systems over a finite field and applications

We study the set of common $\mathbb{F}_q$-rational solutions of "smooth" systems of multivariate symmetric polynomials with coefficients in a finite field $\mathbb{F}_q$. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of $\mathbb{F}_q$ has a "good" geometric behavior. This allows us to obtain precise estimates on the corresponding number of common $\mathbb{F}_q$-rational solutions. In the case of hypersurfaces we are able to improve the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.Comment: 37 pages. arXiv admin note: text overlap with arXiv:1510.03721, arXiv:1807.0805