arXiv subjects
Publications and source records attributed to Herivelto Borges.
We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.
We use a recent characterization of minimal value set polynomials and $q$- Frobenius nonclassical curves to construct curves that generalize the Hermitian curve. The genus $g$ and the number $N$ of $\mathbb{F}_q$-rational points of the curves are computed and, for a special family of these curves, we determine the Weierstrass semigroup at the unique point at infinity. These special curves yield new examples of Castle curves and improve on a previous example of Garcia-Stichtenoth of curves with large ratio $N/g$.
We establish a relation between minimal value set polynomials defined over $\mathbb{F}_q$ and certain $q$-Frobenius nonclassical curves. The connection leads to a characterization of the curves of type $g(y)=f(x)$, whose irreducible components are $q$-Frobenius nonclassical. An immediate consequence will be the realization of rich sources of new $q$-Frobenius nonclassical curves.
We give an explicit characterization of all minimal value set polynomials in $\F_q[x]$ whose set of values is a subfield $\F_{q'}$ of $\F_{q}$. We show that the set of such polynomials, together with the constants of $\F_{q'}$, is an $\F_{q'}$-vector space of dimension $2^{[\F_{q}:\F_{q'}]}$. Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.