arXiv subjects

Risto Korhonen

Publications and source records attributed to Risto Korhonen.

At least 19 recordsLinked to original sources

arXiv2026

n-th Tropical Nevanlinna Theory

In this paper, the tropical Nevanlinna theory is extended for piecewise polynomial continuous functions. By constructing the $n$-th Poisson-Jensen formula, the $n$-th tropical counting, proximity, and characteristic functions are introduced, which have some different properties compared to the classical tropical setting. Then, not only is the $n$-th version of the second main theorem for tropical homogeneous polynomials obtained, but also a tropical second main theorem for ordinary Fermat type polynomials is acquired. Moreover, by estimating the tropical logarithmic derivative with a growth assumption pointwise, a strong equality is proved. This equality illustrates the relationship between $\sum_{i=0}^{m}N(r, 1_{0}\oslash f_{i})$ and the ramification term $N(r, C_{0}(f_{0}, \cdots, f_{m}))$, implying that there is no natural tropical truncated version of the second main theorem for shift operators.

math.AG↗

arXiv2026

On the existence of meromorphic solutions of the complex Schrödinger equation with a q-shift

In this paper, we study the following complex Schrödinger equation with a $q$-difference term: \begin{align}\tag{†}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where $a(z) \not\equiv 0$ is a small meromorphic function with respect to $f(z)$, and all the coefficient functions of $R(z, f(z))$ are also small meromorphic functions with respect to $f(z)$. We assume that $q\in\mathbb{C}\setminus \left \{ 0,-1,1 \right \} $ and that $R(z, f(z))$ is an irreducible rational function in both $f(z)$ and $z$. We obtain some necessary conditions for \eqref{dagger} to have meromorphic solutions of zero order and non-constant entire solutions, respectively. In particular, if $R(z,f(z))$ reduces to a polynomial in $f(z)$ with degree at most 2 and all the coefficients are constant, then under this assumption and without imposing any restrictions on the growth order of $f(z),$ we prove the existence of entire solutions in many cases, study their number, and further investigate the local and global meromorphic solutions to \eqref{dagger}. Additionally, we consider the possible forms of the meromorphic solutions to \eqref{dagger} in certain conditions and examine exponential polynomials as possible solutions of \eqref{dagger}.

math.CV↗

arXiv2026

On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$

In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (Tôhoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen.

math.CV↗

arXiv2025

Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function

We show that for a vanishing period difference operator of a meromorphic function $ f $, there exist the following estimates regarding proximity functions, \[ \lim_{η\to 0} m_η\left(r, \frac{Δ_ηf - aη}{f' - a} \right) = 0 \] and \[ \lim_{r \to \infty} m_η\left(r, \frac{Δ_ηf - aη}{f' - a} \right) = 0, \] where $ Δ_ηf = f(z + η) - f(z) $, and $ |η| $ is less than an arbitrarily small quantity $ α(r) $ in the second limit. Then, under certain assumptions on the growth, restrictions on the period tending to infinity, and on the value distribution of a meromorphic function $ f(z) $, we have \[ m\left(r, \frac{Δ_ωf - aω}{f' - a} \right) = S(r, f'), \] as $ r \to \infty $, outside an exceptional set of finite logarithmic measure. Additionally, we provide an estimate for the angular shift under certain conditions on the shift and the growth. That is, the following Nevanlinna proximity function satisfies \[ m\left(r, \frac{f(e^{iω(r)}z) - f(z)}{f'} \right) = S(r, f), \] outside an exceptional set of finite logarithmic measure. Furthermore, the above estimates yield additional applications, including deficiency relations between $ Δ_ηf $ (or $ Δ_ωf $) and $ f' $, as well as connections between $ η/ω$-separated pair indices and $ δ(0, f') $.

math.CV↗

arXiv2025

Transcendental meromorphic solutions and the complex Schrödinger equation with delay

In this article, we focus on studying the differential-difference equation \[ f'(z) = a(z)f(z+1) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \] where the two nonzero polynomials $ P(z, f(z)) $ and $ Q(z, f(z)) $ in $ f(z) $, with small meromorphic coefficients, are coprime, and $ a(z) $ is a nonzero small meromorphic function of $ f(z) $. This equation includes the complex Schrodinger equation with delay as a special case. If $ f(z) $ is a transcendental meromorphic solution of the equation with subnormal growth, then we derive all possible forms of the equation. Additionally, under these assumptions, we classify these specific forms based on the degrees of $ P(z, f(z)) $ and $ Q(z, f(z)) $ to establish necessary conditions for the existence of transcendental meromorphic solutions. In particular, when the degree of $ P $ minus the degree of $ Q $ is 2, we demonstrate that the equation reduces to a Riccati differential equation. Finally, examples are provided to support our results.

math.CV↗

arXiv2025

On the extension of analytic solutions of first-order difference equations

We will consider first-order difference equations of the form \[ y(z+1) = \frac{λy(z)+a_2(z)y(z)^2+\cdots+a_p(z)y(z)^p}{1 + b_1(z)y(z)+\cdots+b_q(z)y(z)^q}, \] where $λ\in\mathbb{C}\setminus\{0\}$ and the coefficients $a_j(z)$ and $b_k(z)$ are meromorphic. When existence of an analytic solution can be proved for large negative values of $\Re(z)$, the equation determines a unique extension to a global meromorphic solution. In this paper we prove the existence of non-constant meromorphic solutions when the coefficients satisfy $|a_{j}(z)|\leq ν^{|z|}$ and $|b_{k}(z)|\leq ν^{|z|}$ for some $ν<|λ|$ in a half-plane. Furthermore, when a solution exists that is analytic for large positive values of $\Re(z)$, the equation determines a unique extension to a global solution that will generically have algebraic branch points. We analyse a particular constant coefficient equation, $y(z+1)=λy(z)+y(z)^2$, $0<λ<1$, and describe in detail the infinitely-sheeted Riemann surface for such a solution. We also describe solutions with natural boundaries found by Mahler.

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arXiv2024

Askey-Wilson version of Second Main Theorem for holomorphic curves in projective space

In this paper, an Askey-Wilson version of the Wronskian-Casorati determinant $\mathcal{W}(f_{0}, \dots, f_{n})(x)$ for meromorphic functions $f_{0}, \dots, f_{n}$ is introduced to establish an Askey-Wilson version of the general form of the Second Main Theorem in projective space. This improves upon the original Second Main Theorem for the Askey-Wilson operator due to Chiang and Feng. In addition, by taking into account the number of irreducible components of hypersurfaces, an Askey-Wilson version of the Truncated Second Main Theorem for holomorphic curves into projective space with hypersurfaces located in $l$-subgeneral position is obtained.

math.CV↗

arXiv2024

Zero order meromorphic solutions of $q$-difference equations of Malmquist type

We consider the first order $q$-difference equation \begin{equation}\tag† f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(†)$ has a zero order transcendental meromorphic solution, then $(†)$ reduces to a $q$-difference linear or Riccati equation, or to an equation that can be transformed to a $q$-difference Riccati equation. In the autonomous case, explicit meromorphic solutions of $(†)$ are presented. Given that $(†)$ can be transformed into a difference equation, we proceed to discuss the growth of the composite function $f(ω(z))$, where $ω(z)$ is an entire function satisfying $ω(z+1)=qω(z)$, and demonstrate how the proposed difference Painlevé property, as discussed in the literature, applies for $q$-difference equations.

math.CV↗

arXiv2023

A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function

It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{Δ_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $Δ_c f=f(z+c)-f(z)$ and $a,c\in\mathbb{C}$. This estimate implies a lower bound for the Nevanlinna ramification term in terms of the difference operator with an arbitrary shift. As a consequence it follows, for instance, that if $f$ is an entire function of hyper-order $<1$ whose derivative does not attain a value $a\in\mathbb{C}$ often $$N\left(r,\frac{1}{f'-a}\right)=S(r,f),$$ then the finite difference $Δ_c f$ cannot attain the value $ac$ significantly more often $$N\left(r,\frac{1}{Δ_c f-ac}\right)=S(r,f).$$ Additional applications of the estimate above include a new type of a second main theorem, deficiency relations between $Δ_cf$ and $f'$ and new Clunie and Mohon'ko type lemmas.

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arXiv2023

On meromorphic solutions of Malmquist type difference equations

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form $f(z+1)^n=R(z,f)$ $(†)$ that have transcendental meromorphic solutions, where $R(z,f)$ is rational in both arguments. In this paper, we first complete the classification for the case $°_{f}(R(z,f))=n$ of~$(†)$ by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on~$(†)$. Then, we study the relations between $(†)$ and its differential counterpart $(f')^n=R(z,f)$. We show that most autonomous equations, singled out from~$(†)$ with $n=2$, have a natural continuum limit to either the differential Riccati equation $f'=a+f^2$ or the differential equation $(f')^2=a(f^2-τ_1^2)(f^2-τ_2^2)$, where $a\not=0$ and $τ_i$ are constants such that $τ_1^2\not=τ_2^2$. The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.

math.CV↗

arXiv2023

Tropical second main theorem and the Nevanlinna inverse problem

A generalization of the second main theorem of tropical Nevanlinna theory is presented for noncontinuous piecewise linear functions and for tropical hypersurfaces without requiring a growth condition. The method of proof is novel and significantly more straightforward than previously known proofs. The tropical analogue of the Nevanlinna inverse problem is formulated and solved for tropical meromorphic functions and tropical hypersurfaces.

math.AG↗

arXiv2023

A Malmquist--Steinmetz theorem for difference equations

It is shown that if the equation \begin{equation*} f(z+1)^n=R(z,f), \end{equation*} where $R(z,f)$ is rational in both arguments and $°_f(R(z,f))\not=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $°_f(R(z,f))=n$ of the equation above and thus provide a complete difference analogue of Steinmetz' generalization of Malmquist's theorem.

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arXiv2020

Value distribution of q-differences of meromorphic functions in several complex variables

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.

math.CV↗

arXiv2020

Studies of Differences from the point of view of Nevanlinna Theory

This paper consists of three parts. First, we give so far the best condition under which the shift invariance of the counting function, and of the characteristic of a subharmonic function, holds. Second, a difference analogue of logarithmic derivative of a $δ$-subharmonic function is established allowing the case of hyper-order equal to one and minimal hyper-type, which improves the condition of the hyper-order less than one. Finally, we make a careful discussion of a well-known difference equation and give out the possible forms of the equation under a growth condition for the solutions.

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arXiv2018

Mason's theorem with a difference radical

Differential calculus is not a unique way to observe polynomial equations such as $a+b=c$. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials $a$, $b$ and $c$ satisfying the equation above. Then a difference $abc$ theorem for polynomials is proved using a new notion of a radical of a polynomial. Two results on the non-existence of polynomial solutions to difference Fermat type functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider difference Fermat type equations with transcendental entire solutions.

math.CV↗

arXiv2018

A lemma on the difference quotients

Using a new Borel type growth lemma, we extend the difference analogue of the lemma on the logarithmic derivative due to Halburd and Korhonen to the case of meromorphic functions $f(z)$ such that $\log T(r,f)\leq r/(\log r)^{2+ν}$, $ν>0$, for all sufficiently large $r$. The method by Halburd and Korhonen implies an estimate for the lemma on difference quotients, where the exceptional set is of finite logarithmic measure. We show the necessity of this set by proving that it must be of infinite linear measure for meromorphic functions whose deficiency is dependent on the choice of the origin. In addition, we show that there is an infinite sequence of $r$ in the set for which $m(r,f(z+c)/f(z))$ is not small compared to $T(r,f)$ for entire functions constructed by Miles. We also give a discrete version of Borel type growth lemma and use it to extend Halburd's result on first order discrete equations of Malmuist type.

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arXiv2018

Existence of meromorphic solutions of first order difference equations

It is shown that if It is shown that if \begin{equation}\label{abstract_eq} f(z+1)^n=R(z,f),\tag† \end{equation} where $R(z,f)$ is rational in $f$ with meromorphic coefficients and $°_f(R(z,f))=n$, has an admissible meromorphic solution, then either $f$ satisfies a difference linear or Riccati equation with meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if \eqref{abstract_eq}, where the assumption $°_f(R(z,f))=n$ has been discarded, has rational coefficients and a transcendental meromorphic solution $f$ of hyper-order $<1$, then either $f$ satisfies a difference linear or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions which are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz' generalization of Malmquist's theorem.

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arXiv2017

Meromorphic solutions of algebraic difference equations

It is shown that the difference equation \begin{equation}\label{abseq} (Δf(z))^2=A(z)(f(z)f(z+1)-B(z)), \qquad\qquad (1) \end{equation} where $A(z)$ and $B(z)$ are meromorphic functions, possesses a continuous limit to the differential equation \begin{equation}\label{abseq2} (w')^2=A(z)(w^2-1),\qquad\qquad (2) \end{equation} which extends to solutions in certain cases. In addition, if (1) possesses two distinct transcendental meromorphic solutions, it is shown that these solutions satisfy an algebraic relation, and that their growth behaviors are almost same in the sense of Nevanlinna under some conditions. Examples are given to discuss the sharpness of the results obtained. These properties are counterparts of the corresponding results on the algebraic differential equation (2).

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