arXiv subjects

Eder L. Sanchez Quiceno

Publications and source records attributed to Eder L. Sanchez Quiceno.

5 recordsLinked to original sources

arXiv2026

Essential tori associated with links of mixed singularities

We establish a direct connection between the analytic data of weakly isolated mixed singularities and the topology of their associated links. More precisely, we prove that the existence of essential tori, topological information, in the complements of links arising from weakly isolated mixed singularities can be detected directly from properties of the defining mixed polynomial, provided that it is convenient, non-degenerate and $Γ$-nice. Our results provide explicit and computable criteria, expressed purely in terms of the polynomial data, that determine the presence of essential tori in the link exterior. In particular, these criteria yield effective conditions ensuring that such links are non-hyperbolic. This approach provides a new method to extract topological information about link complements without requiring an explicit determination of the link type, thereby establishing a concrete bridge between the analytic structure of mixed polynomials and the geometric topology of their associated links.

math.GT↗

arXiv2026

Topological triviality and link-constancy in deformations of inner Khovanskii non-degenerate maps

For real and mixed polynomial maps $f=(f^1,\dots,f^p)$ satisfying $f(0)=0$, we introduce the notion of Inner Khovanskii Non-Degeneracy (IKND), that generalizes a previous non-degeneracy condition introduced by Wall for complex polynomial functions (J. Reine Angew. Math. 509 (1999), 1-19). We prove that IKND is a sufficient condition ensuring that the link of the singularity of $f$ at the origin is smooth and well-defined. We study one-parameter deformations of an IKND map $f$, given by $F(\boldsymbol{x},\varepsilon)=f(\boldsymbol{x})+θ(\boldsymbol{x},\varepsilon)$, with $ F(0,\varepsilon)=0$. We prove that the deformation is \textit{link-constant} under suitable conditions on $f$ and $θ$, meaning that the ambient isotopy type of the link remains unchanged along the deformation. Furthermore, by employing a strong version of this non-degeneracy, Strong Inner Khovanskii Non-Degeneracy (SIKND), we obtain results on topological triviality. In the final section, we present link-constant deformations for IKND mixed polynomial functions of two variables. We also explore several applications motivated by the recent findings of Araújo dos Santos, Bode, and Sanchez Quiceno (Bull. Braz. Math. Soc. (N.S.) 55 (2024), no. 3, Paper No. 34).

math.AG↗

arXiv2023

Inner and Partial non-degeneracy of mixed functions

Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from $\mathbb{R}^4$ to $\mathbb{R}^2$. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then introduce additional conditions under which strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration.

math.AG↗

arXiv2022

Links of singularities of inner non-degenerate mixed functions

We introduce the notion of a (strongly) inner non-degenerate mixed function $f:\mathbb{C}^2\to\mathbb{C}$. We show that inner non-degenerate mixed polynomials have weakly isolated singularities and strongly inner non-degenerate mixed polynomials have isolated singularities. Furthermore, under one additional assumption, which we call "niceness", the links of these singularities can be completely characterized in terms of the Newton boundary of $f$. In particular, adding terms above the Newton boundary does not affect the topology of the link. This allows us to define an infinite family of real algebraic links in the 3-sphere.

math.GT↗

arXiv2022

On real algebraic links in the 3-sphere associated with mixed polynomials

In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the $3$-sphere. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti-Shiota conjecture, which state that any fibered link on the $3$-sphere is a real algebraic link.

math.GT↗