arXiv subjects

Erik A. R. Mendoza

Publications and source records attributed to Erik A. R. Mendoza.

7 recordsLinked to original sources

arXiv2025

On gap sets in arbitrary Kummer extensions of $K(x)$

Let $K$ be an algebraically closed field, and let $F/K(x)$ be a Kummer extension of function fields of genus $g$. We provide a compact and explicit description of the gap set $G(Q)$ at any totally ramified place $Q$ of the extension $F/K(x)$. As a consequence, we deduce structural properties of the Weierstrass semigroup $H(Q)$; in particular, we determine a generating set for $H(Q)$, and we characterize its symmetry in certain cases. We also generalize a formula due to Towse that describes the asymptotic behavior of the sum of the Weierstrass weights at all totally ramified places of the extension $F/K(x)$ relative to $g^3-g$.

math.AG↗

arXiv2025

On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$

In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of function field $\mathbb{F}_q(x)$. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$, which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.

math.AG↗

arXiv2023

The Set of Pure Gaps at Several Rational Places in Function Fields

In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field $F$ over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters.

cs.IT↗

arXiv2023

Weierstrass Semigroup, Pure Gaps and Codes on Function Fields

We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-α_i)^{λ_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where $α_1, \dots, α_r\in K$ are pairwise distinct elements, and $\gcd(m, \sum_{i=1}^{r}λ_i)=1$. For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. We apply these results to construct algebraic geometry codes over certain function fields with many rational places.

math.AG↗

arXiv2023

Complete set of Pure Gaps in Function Fields

In this work, we provide a way to completely determine the set of pure gaps $G_0(P_1, P_2)$ at two rational places $P_1, P_2$ in a function field $F$ over a finite field $\mathbb{F}_q$, and its cardinality. Furthermore, we given a bound for the cardinality of the set $G_0(P_1, P_2)$ which is better, in some cases, than the generic bound given by Homma and Kim. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the $GK$ function field and Kummer extensions.

math.NT↗

arXiv2023

On Kummer extensions with one place at infinity

Let $K$ be the algebraic closure of $\mathbb{F}_{q}$. We provide an explicit description of the Weierstrass semigroup $H(Q_\infty)$ at the only place at infinity $Q_{\infty}$ of the curve $\mathcal{X}$ defined by the Kummer extension with equation $y^m=f(x)$, where $f(x)\in K[x]$ is a polynomial satisfying $\gcd (m, \text{deg} f)=1$. As a consequence, we determine the Frobenius number and the multiplicity of $H(Q_{\infty})$ in some cases, and we discuss sufficient conditions for the Weierstrass semigroup $H(Q_{\infty})$ to be symmetric. Finally, we characterize certain maximal Castle curves of type $(\mathcal{X}, Q_{\infty})$.

math.AG↗

arXiv2021

Reciprocal polynomials and curves with many points over a finite field

Let $\mathbb F_{q^2}$ be the finite field with $q^2$ elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over $\mathbb F_{q^2}$ with many rational points. The curves constructed are Kummer covers or fibre products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves.

math.NT↗