481 research outputs found
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 CientĂficas y TĂ©cnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Matera, Guillermo. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; 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 CientĂficas y TĂ©cnicas; Argentin
Smooth symmetric systems over a finite field and applications
We study the set of common -rational solutions of "smooth"
systems of multivariate symmetric polynomials with coefficients in a finite
field . We show that, under certain conditions, the set of common
solutions of such polynomial systems over the algebraic closure of
has a "good" geometric behavior. This allows us to obtain
precise estimates on the corresponding number of common -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
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