Classifying (near)-Belyi maps with Five Exceptional Points

Tsukuba Journal of Mathematics

In this paper, we investigate the relationship between di¤erential polynomials and meromorphic functions of finite order of some second order linear di¤erential equations with meromorphic coe‰cients. We obtain some precise estimates. 1 Introduction and Statement of Results Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna's value distribution theory (see [9], [13]). In addition, we will use lð f Þ and lð1=f Þ to denote respectively the exponents of convergence of the zero-sequence and the polesequence of a meromorphic function f , rð f Þ to denote the order of growth of f , lð f Þ and lð1=f Þ to denote respectively the exponents of convergence of the sequence of distinct zeros and distinct poles of f. Consider the second order linear di¤erential equation f 00 þ A 1 ðzÞe PðzÞ f 0 þ A 0 ðzÞe QðzÞ f ¼ 0; ð1:1Þ where PðzÞ, QðzÞ are nonconstant polynomials, A 1 ðzÞ, A 0 ðzÞ ðD 0Þ are entire functions such that rðA 1 Þ < deg PðzÞ, rðA 0 Þ < deg QðzÞ. In [11], Ki-Ho Kwon has investigated the hyper order of solutions of (1.1) when deg PðzÞ ¼ deg QðzÞ. Gundersen showed in [7, p. 419] that if deg PðzÞ 0 deg QðzÞ, then every nonconstant solution of (1.1) is of infinite order. If deg PðzÞ ¼ deg QðzÞ, then (1.1)

We characterize generalized hypergeometric series that solve a linear differential equation with polynomial coefficients at an ordinary point of the equation, and show that these solutions remain hypergeometric at any other ordinary point. Therefore to find all generalized hypergeometric series solutions, it suffices to look at a finite number of points: all the singular points, and a single, arbitrarily chosen ordinary point.

Finite Fields and Their Applications, 2003

We study some classes of equations with Carlitz derivatives for F q-linear functions, which are the natural function field counterparts of linear ordinary differential equations with a regular singularity. In particular, an analog of the equation for the power function, the Fuchs and Euler type equations, and Thakur's hypergeometric equation are considered. Some properties of the above equations are similar to the classical case while others are different. For example, a simple model equation shows a possibility of existence of a non-trivial continuous locally analytic F q-linear solution which vanishes on an open neighbourhood of the initial point.

Journal of Differential Equations, 2010

An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some local chart by a differential equation with algebraic coefficients. We show that there exist at least countably many saddle-node singularities of the complex plane that are not algebraizable.

Confluentes Mathematici, 2019

On the solutions of the universal differential equation with three regular singularities (On solutions of 3) Tome 11, n o 2 (2019), p. 25-64. <http://cml.centre-mersenne.org/item?id=CML_2019__11_2_25_0> © Les auteurs et Confluentes Mathematici, 2019. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence Licence Internationale d'Attibution Creative Commons BY 4.0 https://creativecommons.org/licenses/by/4.0 L'accès aux articles de la revue « Confluentes Mathematici » (http://cml.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://cml.centre-mersenne. org/legal/). Confluentes Mathematici est membre du Centre Mersenne pour l'édition scientifique ouverte http://www.centre-mersenne.org/

The ANZIAM Journal, 2002

We have finally obtained for each of the 6 Painlevés an expression of z, w, w′ that behaves as 1/(z − Z0) + O(1) at each kind of movable singular point. This expression is polynomial in w′ (at most quadratic), and rational in w and z. After it is integrated and exponentiated it yields a function that has a simple zero at each of the singular points.

Journal of Differential Equations, 2017

In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

Opuscula Mathematica, 2017

In this paper, we investigate the growth of meromorphic solutions of the linear differential equation f (k) + h k−1 (z)e P k−1 (z) f (k−1) +. .. + h0(z)e P 0 (z) f = 0, where k ≥ 2 is an integer, Pj(z) (j = 0, 1,. .. , k − 1) are nonconstant polynomials and hj(z) are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations.

This paper deals with singularities of nonconfluent hypergeometric functions in several complex variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of the amoebas and the Newton polytopes of their defining polynomials. In particular, we show that the amoebas of classical discriminantal hypersurfaces are solid, that is, they possess the minimal number of complement components.

Salahaddin University-Erbil, 2016

In this thesis, we presented some Nonstandard concepts to study the analyticity near the singularity. We analyzed and proved the Existence and Uniqueness Theorems for first order ordinary differential equations in the subset of the monad of the initial standard point. Then the solutions of second order ordinary differential equation (Legendre Equation) are found around the singularity in the monad of zero by using power series method with suitable transformations for singular points. Additionally, we wrote the Legendre polynomial on the form of Mehler-Dirichlet Integral formula to find its solutions near the singularity. Furthermore, Nonstandard analysis tools are successfully applied to find a Nonstandard analytic solution for the first order differential equation near singularity in the following cases:  The differential equation with one of the coefficients is unlimited. The general first order ODE in this case were of the form: dy dx = f (x, y) = 1 P(x, y) , where P(x, y) is a polynomial in x and y such that P(x, y) = a1,0x + a0,1y + a1,1xy + a0,2x2 +


+ aw1,w2xw1yw2, where ai,j is limited for each i, j  0 and w1, w2 are unlimited with 1 wp /2 z 􀀀 Microhal(0) and z is an infinitesimal, for p The differential coefficients are infinitesimals or unlimited and the first order differential equation have the form: dy ω1,ω2 i j xdx = f (x, y) = αx + βy + ∑ i,j=0 i+j≥2 ai,jx y , where ai,j is limited for each i, j ≥ 0 and ω₁, ω₂ are unlimited with 1 , 1 ∈ (ζ − Microhal(0))ᶜ. ω₁ ω₂ • The differential equation have irreducible differential Form. M(x, y)dx + N(x, y)dy = 0, where M(x, y) and N(x, y) is a power series in x and y. Therefore, it is of the form: ω ,ω j=0 2 ai,jxiyj)dx + (aˆx + + ω3,ω 4 j , i,j=0 i+j≥2 where 4 is unlimited with 1 / Microhal 0 and the ωk coefficients are limited. ∏ k=1 ∈ ζ − ( ) • Analyticity of system of differential equation in the monad of singu- larity. We considered the following system: dx₁ φ₁ dx₂ = φ₂ dxn = · · · = , where each φ₁, φ₂, · · · , φn is a power series in x₁, x₂, · · · , xn and n is standard natural number. Finally, we studied in details the idea of the proof of Painleve´’s Theorem in Nonstandard analysis. VII