arXiv subjects

Sergio Ricardo Zapata Ceballos

Publications and source records attributed to Sergio Ricardo Zapata Ceballos.

3 recordsLinked to original sources

arXiv2026

On the Squarefree Values of Degree-$2q$ Polynomials

In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.

math.NT↗

arXiv2025

Unit lattices of $D_4$-quartic number fields with signature $(2,1)$

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$.

math.NT↗

arXiv2025

On the proportion of metric matroids whose Jacobians have nontrivial $\mathbf{p}$-torsion

We study the proportion of metric matroids whose Jacobians have nontrivial $p$-torsion. We establish a correspondence between these Jacobians and the $\mathbb{F}_p$-rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to $1/p$.

math.CO↗