arXiv subjects

Jesús Muciño-Raymundo

Publications and source records attributed to Jesús Muciño-Raymundo.

8 recordsLinked to original sources

arXiv2025

Tessellations of rational complex functions and the Riemann's existence theorem

A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T.

math.CV↗

arXiv2025

Abelian integrals for polynomials with trivial global monodromy on $\mathbb{C}^2$

We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon ω=0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $ω$ is a non-exact polynomial 1-form of degree $n$. In order to study these perturbed differential equations, the associated Abelian integrals $I(c)=\int_{γ(c)} ω$ are valuable tools. We assume that the polynomials $H$ are primitive with trivial global monodromy. For these polynomials, W. D. Neumann and P. Norbury provided a classification in three large families, up to algebraic equivalence. The knowledge of these families allows us to prove as first main result, that the respective Abelian integrals $I(c)$ are polynomial functions of the variable $c$, and to find sharp explicit upper bounds for the number of their zeros. The bounds depend on $m$, $n$ and the number of the generators of the fundamental group of the generic fibers of $H$. These upper bounds works for several new families of infinitesimal perturbations of Hamiltonian differential equations. Under trivial global monodromy, there exist canonical global generators $BC(H)= \{ γ_{\tt i}(c)\}$ of the fundamental groups for all the generic fibers of $H$, which are complex cycles of $dH=0$. As second main result; we compute the number of complex limit cycles of $dH+ \varepsilonω=0$ which originate from complex cycles in $BC(H)$. Several accurate examples are provided.

math.DS↗

arXiv2024

Geometry of transcendental singularities of complex analytic functions and vector fields

On Riemann surfaces $M$, there exists a canonical correspondence between a possibly multivalued function $Ψ_X$ whose differential is single valued ($i.e.$ an additively automorphic singular complex analytic function) and a vector field $X$. From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities and accumulation points of the above. The theory of singularities of the inverse function $Ψ_X^{-1}$ is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of $Ψ_X^{-1}$ is either algebraic, logarithmic or arises from a zero with nonzero residue of $X$. Relationships between analytical properties of $Ψ_X$, singularities of $Ψ_X^{-1}$ and singularities of $X$ are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of $Ψ_X^{-1}$ are studied. As applications we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field $X$, a geometric characterization of the incomplete real trajectories of a vector field $X$, and a description of the singularities of the vector field associated to the Riemann $ξ$ function.

math.CV↗

arXiv2022

Integrability and adapted complex structures to smooth vector fields on the plane

Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular complex analytic vector fields $\mathbb{X}$ and smooth vector fields $X$. Our approximation route studies three integrability notions for real smooth vector fields $X$ with singularities on the plane or the sphere. The first notion is related to Cauchy-Riemann equations, we say that a vector field $X$ admits an adapted complex structure $J$ if there exists a singular complex analytic vector field $X$ on the plane provided with this complex structure, such that $X$ is the real part of $\mathbb{X}$. The second integrability notion for $X$ is the existence of a first integral $f$, smooth and having non vanishing differential outside of the singularities of $X$. A third concept is that $X$ admits a global flow box map outside of its singularities, i.e. the vector field $X$ is a lift of the trivial horizontal vector field, under a diffeomorphism. We study the relation between the three notions. Topological obstructions (local and global) to the three integrability notions are described. A construction of singular complex analytic vector fields X using canonical invariant regions is provided.

math.DS↗

arXiv2022

Plane polynomials and Hamiltonian vector fields determined by their singular points

Let $Σ(f)$ be critical points of a polynomial $f \in \mathbb{K}[x,y]$ in the plane $\mathbb{K}^2$, where $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$. Our goal is to study the critical point map $\mathfrak{S}_d$, by sending polynomials $f$ of degree $d$ to their critical points $Σ(f)$ . Very roughly speaking, a polynomial $f$ is essentially determined when any other $g$ sharing the critical points of $f$ satisfies that $f= λg$; here both are polynomials of at most degree $d$, $λ\in \mathbb{K}^*$. In order to describe the degree $d$ essentially determined polynomials, a computation of the required number of isolated critical points $δ(d)$ is provided. A dichotomy appears for the values of $δ(d)$; depending on a certain parity the space of essentially determined polynomials is an open or closed Zariski set. We compute the map $\mathfrak{S}_3$, describing under what conditions a configuration of four points leads to a degree three essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree three non essential determined polynomials. The quotient space of essentially determined polynomials of degree three up to the action of the affine group $\hbox{Aff}(\mathbb{K}^2)$ determines a singular surface over $\mathbb{K}$.

math.AG↗

arXiv2021

Dynamics of singular complex analytic vector fields with essential singularities II

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P, E\in\mathbb{C}[z]\right\}$, where $P$ is monic, $deg(P)=r$, $deg(E)=d$, $r+d\geq 1$, have a finite number of poles on the complex plane and an isolated essential singularity at infinity (for $d\geq 1$). Our aim is to describe geometrically $X$, particularly the singularity at infinity. We use the natural one to one correspondence between $X$, a global singular analytic distinguished parameter $Ψ_X(z)=\int^z P(ζ) e^{-E(ζ)}dζ$, and the Riemann surface ${\mathcal R}_X$ of this distinguished parameter. We introduce $(r,d)$-configuration trees which are weighted directed rooted trees. An $(r,d)$-configuration tree completely encodes the Riemann surface ${\mathcal R}_X$ and the singular flat metric associated on ${\mathcal R}_X$. The $(r,d)$-configuration trees provide "parameters" for the complex manifold ${\mathscr E}(r,d)$, which give explicit geometrical and dynamical information; a valuable tool for the analytic description of $X\in{\mathscr E}(r,d)$. Furthermore, given $X$, the phase portrait of the associated real vector field $Re(X)$ on the Riemann sphere is decomposed into $Re(X)$-invariant components: half planes and finite height strips. The germ of $X$ at infinity is described as a combinatorial word (consisting of hyperbolic, elliptic, parabolic and entire angular sectors having the point at infinity of $\widehat{\mathbb C}_z$ as center). The structural stability, under perturbation in ${\mathscr E}(r,d)$, of the phase portrait of $Re(X)$ is characterized by using the $(r,d)$-configuration trees. We provide explicit conditions, in terms of $r$ and $d$, as to when the number of topologically equivalent phase portraits of $Re(X)$ is unbounded.

math.DS↗

arXiv2020

On the geometry, flows and visualization of singular complex analytic vector fields on Riemann surfaces

Motivated by the wild behavior of isolated essential singularities in complex analysis, we study singular complex analytic vector fields $X$ on arbitrary Riemann surfaces $M$. By vector field singularities we understand zeros, poles, isolated essential singularities and accumulation points of the above kind. In this framework, a singular analytic vector field $X$ has canonically associated; a 1-form, a quadratic differential, a flat metric (with a geodesic foliation), a global distinguished parameter or $\mathbb{C}$-flow box $Ψ_X$, a Newton map $Φ_X$, and a Riemann surface $\mathcal{R}_X$ arising from the maximal $\mathbb{C}$-flow of $X$. We show that every singular complex analytic vector field $X$ on a Riemann surface is in fact both a global pullback of the constant vector field under $Ψ_X$ and of the radial vector field on the sphere under $Φ_X$. As a result of independent interest, we show that the maximal analytic continuation of the a local $\mathbb{C}$-flow of $X$ is univalued on the Riemann surface $\mathcal{R}_{X} \subset M \times \mathbb{C}_t$, where $\mathcal{R}_X$ is the graph of $Ψ_{X}$. Furthermore we explore the geometry of singular complex analytic vector fields and present a geometrical method that enables us to obtain the solution, without numerical integration, to the differential equation that provides the $\mathbb{C}$-flow of the vector field. We discuss the theory behind the method, its implementation, comparison with some integration-based techniques, as well as examples of the visualization of complex vector fields on the plane, sphere and torus. Applications to visualization of complex valued functions is discussed including some advantages between other methods.

math.DS↗

arXiv2019

Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

We consider the family $\mathcal{E}(s,r,d)=\Big\{ X(z)=\frac{Q(z)}{P(z)}\ e^{E(z)} \frac{\partial}{\partial z} \Big\},$ with $Q, P, E$ polynomials, deg${Q}=s$, deg$(P)=r$ and deg$(E)=d$, of singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}$. For $d\geq1$, $X\in\mathcal{E}(s,r,d)$ has $s$ zeros and $r$ poles on the complex plane and an essential singularity at infinity. Using the pullback action of the affine group $Aut(\mathbb{C})$ and the divisors for $X$, we calculate the isotropy groups $Aut(\mathbb{C})_{X}$ and the discrete symmetries for $X\in\mathcal{E}(s,r,d)$. Each subfamily $\mathcal{E}(s,r,d)_{id}$, of those $X$ with trivial isotropy group in $Aut(\mathbb{C})$, is endowed with a holomorphic trivial principal $Aut(\mathbb{C})$-bundle structure. Necessary and sufficient conditions in order to ensure the equality $\mathcal{E}(s,r,d)=\mathcal{E}(s,r,d)_{id}$ and those $X\in\mathcal{E}(s,r,d)$ with non-trivial isotropy are realized. Explicit global normal forms for $X\in\mathcal{E}(s,r,d)$ are presented. A natural dictionary between vector fields, 1-forms, quadratic differentials and functions is extended to include the presence of non-trivial discrete symmetries $Γ<Aut(\mathbb{C})$.

math.DS↗