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Tingbin Cao

Publications and source records attributed to Tingbin Cao.

15 recordsLinked to original sources
arXiv2026

Tropical Cartan's second main theorem for hyperplanes in general position

We prove a tropical analogue of Cartan's second main theorem for holomorphic curves intersecting hyperplanes in general position--a setting that was not fully resolved by previous tropical Nevanlinna theory. Two versions are obtained. The first (Theorem 1.7) requires subnormal growth and involves the tropical Casorati determinant. The second and main version (Theorem 1.9) is completely free of growth conditions and exceptional sets; it replaces the Casorati term by the sum of the counting functions of the curve's components, yielding an inequality valid for every r. The proof uses a tropical Cramer theorem, bypassing the logarithmic derivative lemma. This improves upon previous results by Korhonen-Tohge and Cao-Zheng, where the coefficient could be suboptimal even under the general position hypothesis. We also clarify the relation between different notions of linear independence, and present the first counterexample to the truncated second main theorem in the tropical setting (Example 5.4).

math.AG↗
arXiv2025

On the Fermat-type partial differential-difference equations on $\mathbb{C}^n$

Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and $\overline{f}(z)=f(z_{1}+c_{1}, \ldots, z_{n}+c_{n})$, where $q_{t_j}$ ($j=1,2, \cdots, s\le n$) are non-zero polynomials on $\mathbb{C}^{n}$ and $c=(c_{1}, \ldots, c_{n})\in \mathbb{C}^{n}\setminus\{0\}$. We show the structures of all entire solutions to the non-linear partial differential-difference equation $$(p_{1} L(f)+p_{2}\overline{f}+p_5 f)^{2}+(p_{3}L(f)+p_{4}\overline{f}+p_6 f)^{2}=p.$$ The partial differential-difference equation is called a Fermat-type partial differential-difference equation (PDDE). Further, we find many sufficient conditions and/or necessary conditions for the existence, as well as the concrete representations, of entire solutions to the Fermat-type PDDE. We also demonstrate several examples on $\mathbb{C}^2$ with non-constant coefficients to verify that all representations in our theorems exist and are accurate and that the entire solutions to the Fermat-type PDDEs could have finite or infinite growth order. Our theorems unify and extend previous results (see, e.g., [2, 3, 10, 12, 32]).

math.CV↗
arXiv2025

Tropical Nevanlinna theory of several variables

The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher-dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically nondegenerate tropical holomorphic maps $f$ with subnormal growth from $\mathbb{R}^n$ into tropical projective space $\mathbb{TP}^{m}$ intersecting tropical hypersurfaces $\{V_{P_j}\}_{j=1}^{q}$ with degree $d_{j},$ we then obtain the Second Main Theorem $$\|\,\,\, (q-M-1-\lambda)T_f(r) \leq \sum_{j=M+2}^q \tfrac{1}{d_j}N(r,1_{\mathbb{T}} \oslash P_j \circ f) + o(T_f(r)),$$ where $d=lcd(d_{1}, \ldots, d_{q})$ and $M=(_d^{m+d})-1.$

math.AG↗
arXiv2025

Transcendental meromorphic solutions and the complex Schr\"{o}dinger 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↗
arXiv2022

Riemann-Hurwitz theorem and second main theorem for harmonic morphisms on graphs or metrized complexes

In this article, we mainly obtain the Riemann-Hurwitz theorems for harmonic morphisms on (vertex-weighted) metric graphs or metrized complexes of algebraic curves, inspired of the recent work on harmonic morphisms of graphs or metrized complexes due to many researchers. By making use of these Riemann-Hurwitz theorems, we then systematically establish the second main theorems for harmonic morphisms on finite graphs, vertex-weighted graphs, (vertex-weighted) metric graphs or metrized complexes of algebraic curves, from the viewpoint of Nevanlinna theory.

math.AG↗
arXiv2022

A generalized second main theorem for closed subschemes

Let $Y_{1}, \ldots, Y_{q}$ be closed subschemes which are located in $\ell$-subgeneral position with index $\kappa$ in a complex projective variety $X$ of dimension $n.$ Let $A$ be an ample Cartier divisor on $X.$ We obtain that if a holomorphic curve $f:\mathbb C \to X$ is Zariski-dense, then for every $\epsilon >0,$ \begin{eqnarray*} \sum^{q}_{j=1}\epsilon_{Y_{j}}(A)m_{f}(r,Y_{j})\leq_{exc} \left(\frac{(\ell-n+\kappa)(n+1)}{\kappa}+\epsilon\right)T_{f,A}(r). \end{eqnarray*}This generalizes the second main theorems for general position case due to Heier-Levin [AM J. Math. 143(2021), no. 1, 213-226] and subgeneral position case due to He-Ru [J. Number Theory 229(2021), 125-141]. In particular, whenever all the $Y_j$ are reduced to Cartier divisors, we also give a second main theorem with the distributive constant. The corresponding Schmidt's subspace theorem for closed subschemes in Diophantine approximation is also given.

math.AG↗
arXiv2021

Meromorphic solutions of delay differential equations related to logistic type and generalizations

Let $\{b_{j}\}_{j=1}^{k}$ be meromorphic functions, and let $w$ be admissible meromorphic solutions of delay differential equation $$w'(z)=w(z)\left[\frac{P(z, w(z))}{Q(z,w(z))}+\sum_{j=1}^{k}b_{j}(z)w(z-c_{j})\right]$$ with distinct delays $c_{1}, \ldots, c_{k}\in\mathbb{C}\setminus\{0\},$ where the two nonzero polynomials $P(z, w(z))$ and $Q(z, w(z))$ in $w$ with meromorphic coefficients are prime each other. We obtain that if $\limsup_{r\rightarrow\infty}\frac{\log T(r, w)}{r}=0,$ then $$deg_{w}(P/Q)\leq k+2.$$ Furthermore, if $Q(z, w(z))$ has at least one nonzero root, then $deg_{w}(P)=deg_{w}(Q)+1\leq k+2;$ if all roots of $Q(z, w(z))$ are nonzero, then $deg_{w}(P)=deg_{w}(Q)+1\leq k+1;$ if $deg_{w}(Q)=0,$ then $deg_{w}(P)\leq 1.$\par In particular, whenever $deg_{w}(Q)=0$ and $deg_{w}(P)\leq 1$ and without the growth condition, any admissible meromorphic solution of the above delay differential equation (called Lenhart-Travis' type logistic delay differential equation) with reduced form can not be an entire function $w$ satisfying $\overline{N}(r, \frac{1}{w})=O(N(r, \frac{1}{w}));$ while if all coefficients are rational functions, then the condition $\overline{N}(r, \frac{1}{w})=O(N(r, \frac{1}{w}))$ can be omitted. Furthermore, any admissible meromorphic solution of the logistic delay differential equation (that is, for the simplest special case where $k=1$ and $deg_{w}(P/Q)=0$ ) satisfies that $N(r,w)$ and $T(r, w)$ have the same growth category. Some examples support our results.

math.CV↗
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

Schmidt's subspace theorem for moving hypersurface targets in subgeneral position with index in algebraic variety

Recently, Xie-Cao [15] obtained a Second Main Theorem for moving hypersurfaces located in subgeneral position with index which is extended the result of Ru [11]. By using some methods due to Son-Tan-Thin [13], Quang [9] and Xie-Cao [15], we shall give a Schmidt's Subspace Theorem for moving hypersurface targets in subgeneral position with index intersecting algebraic variety. Our result is a extension the Schmidt's Subspace Theorem due to Son-Tan-Thin [13] and Quang [9].

math.NT↗
arXiv2019

Second main theorem for holomorphic curves into algebraic varieties intersecting moving hypersurfaces targets

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru's second main theorem for moving hypersurfaces targets located in subgeneral position with index.

math.CV↗
arXiv2019

Nevanlinna theory for Jackson difference operators and entire solutions of q-difference equations

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.

math.CV↗
arXiv2018

Second main theorem with tropical hypersurfaces and defect relation

The tropical Nevanlinna theory is Nevanlinna theory for tropical functions or maps over the max-plux semiring by using the approach of complex analysis. The main purpose of this paper is to study the second main theorem with tropical hypersurfaces into tropical projective spaces and give a defect relation which can be regarded as a tropical version of the Shiffman's conjecture. On the one hand, our second main theorem improves and extends the tropical Cartan's second main theorem due to Korhonen and Tohge [Advances Math. 298(2016), 693-725]. The growth of tropical holomorphic curve is also improved to $\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ (rather than just hyperorder strictly less than one) by obtaining an improvement of tropical logarithmic derivative lemma. On the other hand, we obtain a new version of tropical Nevanlinna's second main theorem which is different from the tropical Nevanlinna's second main theorem obtained by Laine and Tohge [Proc. London Math. Soc. 102(2011), 883-922]. The new version of the tropical Nevanlinna's second main theorem implies an interesting defect relation that $\delta_{f}(a)=0$ holds for a nonconstant tropical meromorphic function $f$ with $\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ and any $a\in\mathbb{R}$ such that $f\oplus a\not\equiv a.$

math.CV↗
arXiv2016

A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables

Let $c\in \mathbb{C}^{m},$ $f:\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over the field $\mathcal{P}_{c}$ of $c$-periodic meromorphic functions in $\mathbb{C}^{m}$, and let $H_{j}$ $(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position of $\mathbb{P}^{n}(\mathbb{C}).$ We prove a new version of the second main theorem for meromorphic mappings of hyperorder strictly less than one without truncated multiplicity by considering the Casorati determinant of $f$ instead of its Wronskian determinant. As its applications, we obtain a defect relation, a uniqueness theorem and a difference analogue of generalized Picard theorem.

math.CV↗