The Experts below are selected from a list of 324 Experts worldwide ranked by ideXlab platform
Min Ru - One of the best experts on this subject based on the ideXlab platform.
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The Second Main Theorem in the hyperbolic case
Mathematische Annalen, 2019Co-Authors: Min Ru, Nessim SibonyAbstract:We develop Nevanlinna’s theory for a class of holomorphic maps when the source is a disc. Such maps appear in the theory of foliations by Riemann Surfaces.
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A Cartan’s Second Main Theorem Approach in Nevanlinna Theory
Acta Mathematica Sinica, 2018Co-Authors: Min RuAbstract:In 2002, in the paper entitled “A subspace Theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace Theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original Theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.
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a cartan s second Main Theorem approach in nevanlinna theory
Acta Mathematica Sinica, 2018Co-Authors: Min RuAbstract:In 2002, in the paper entitled “A subspace Theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace Theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original Theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.
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An improvement of Chen-Ru-Yan’s degenerated second Main Theorem
Science China-mathematics, 2015Co-Authors: Min RuAbstract:We give an improvement for the second Main Theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al. (2012). An explicit estimate for the truncation level is also obtained in the projective normal case.
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an improvement of chen ru yan s degenerated second Main Theorem
Science China-mathematics, 2015Co-Authors: Min RuAbstract:We give an improvement for the second Main Theorem of algebraically non-degenerate holomorphic curves into a complex projective variety V intersecting hypersurfaces in subgeneral position, obtained by Chen et al. (2012). An explicit estimate for the truncation level is also obtained in the projective normal case.
Risto Korhonen - One of the best experts on this subject based on the ideXlab platform.
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a new version of the second Main Theorem for meromorphic mappings intersecting hyperplanes in several complex variables
Journal of Mathematical Analysis and Applications, 2016Co-Authors: Risto KorhonenAbstract:Abstract Let c ∈ C m , f : C m → P n ( C ) be a linearly nondegenerate meromorphic mapping over the field P c of c-periodic meromorphic functions in C m , and let H j ( 1 ≤ j ≤ q ) be q ( > 2 N − n + 1 ) hyperplanes in N-subgeneral position of P n ( 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.
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difference analogue of cartan s second Main Theorem for slowly moving periodic targets
Annales Academiae Scientiarum Fennicae. Mathematica, 2016Co-Authors: Risto Korhonen, Nan Li, Kazuya TohgeAbstract:We extend the difference analogue of Cartan’s second Main Theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second Main Theorem. As applications, we obtain a new Picard type Theorem and difference analogues of the deficiency relation for holomorphic curves.
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a new version of the second Main Theorem for meromorphic mappings intersecting hyperplanes in several complex variables
arXiv: Complex Variables, 2016Co-Authors: Risto KorhonenAbstract: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.
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second Main Theorem in the tropical projective space
arXiv: Complex Variables, 2014Co-Authors: Risto Korhonen, Kazuya TohgeAbstract:Tropical Nevanlinna theory, introduced by Halburd and Southall as a tool to analyze integrability of ultra-discrete equations, studies the growth and complexity of continuous piecewise linear real functions. The purpose of this paper is to extend tropical Nevanlinna theory to n-dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves, and by proving a tropical analogue of Cartan's second Main Theorem. It is also shown that in the 1-dimensional case this result implies a known tropical second Main Theorem due to Laine and Tohge.
Si Duc Quang - One of the best experts on this subject based on the ideXlab platform.
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Degeneracy second Main Theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications
International Journal of Mathematics, 2020Co-Authors: Si Duc QuangAbstract:In this paper, we establish a new second Main Theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second Main Theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence Theorem for meromorphic mappings sharing moving hypersurfaces is given.
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quantitative subspace Theorem and general form of second Main Theorem for higher degree polynomials
arXiv: Number Theory, 2018Co-Authors: Si Duc QuangAbstract:This paper deals with the quantitative Schmidt's subspace Theorem and the general from of the second Main Theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace Theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second Main Theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.
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Degeneracy second Main Theorems for meromorphic mappings into projective varieties with hypersurfaces
Transactions of the American Mathematical Society, 2017Co-Authors: Si Duc QuangAbstract:In this paper, we establish a second Main Theorem for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with truncated counting functions. Let $f$ be an algebraically nondegenerate meromorphic mapping from a smooth projective subvariety $V\subset\mathbb P^n(\mathbb C)$ of dimension $k\ge 1$ and let $Q_1,...,Q_q$ be $q$ hypersurfaces in $\mathbb P^n(\mathbb C)$ of degree $d_i$, located in $N-$subgeneral position in $V$. We will prove that, for every $\epsilon >0$, there exists a positive integer $M$ such that $$||\ (q-(N-k+1)(k+1)-\epsilon) T_f(r)\le\sum_{i=1}^q\frac{1}{d_i}N^{[M]}(r,f^*Q_i)+o(T_f(r)).$$ Our result is an extension of the classical second Main Theorem of H. Cartan, also is a generalization of the recent second Main Theorem of M. Ru and improves some recent results.
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second Main Theorem and unicity of meromorphic mappings for hypersurfaces in projective varieties
Acta Mathematica Vietnamica, 2017Co-Authors: Si Duc Quang, Do Phuong AnAbstract:Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\). A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i1<⋯
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Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties
Acta Mathematica Vietnamica, 2016Co-Authors: Si Duc Quang, Phuong AnAbstract:Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\). A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i1
Julie Tzuyueh Wang - One of the best experts on this subject based on the ideXlab platform.
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truncated second Main Theorem with moving targets
Transactions of the American Mathematical Society, 2003Co-Authors: Min Ru, Julie Tzuyueh WangAbstract:We prove a truncated Second Main Theorem for holomorphic curves intersecting a finite set of moving or fixed hyperplanes. The set of hyperplanes is assumed to be non-degenerate. Previously only general position or subgeneral position was considered.
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the truncated second Main Theorem of function fields
Journal of Number Theory, 1996Co-Authors: Julie Tzuyueh WangAbstract:Abstract We formulate some results of function fields which are analogous to Cartan's Second Main Theorem with truncated counting functions, Cartan's conjecture, and the generalized Picard Theorem. We also give a generalization of function fields version of abc -conjecture due to Mason, Voloch, Brownawell, and Masser. A result of the finiteness of the S -integral points of function fields of characteristic 0 in the complement of 2 n +1 hyperplanes with constant coefficients and in general position is obtained in this paper.
Kazuya Tohge - One of the best experts on this subject based on the ideXlab platform.
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difference analogue of cartan s second Main Theorem for slowly moving periodic targets
Annales Academiae Scientiarum Fennicae. Mathematica, 2016Co-Authors: Risto Korhonen, Nan Li, Kazuya TohgeAbstract:We extend the difference analogue of Cartan’s second Main Theorem for the case of slowly moving periodic hyperplanes, and introduce two different natural ways to find a difference analogue of the truncated second Main Theorem. As applications, we obtain a new Picard type Theorem and difference analogues of the deficiency relation for holomorphic curves.
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second Main Theorem in the tropical projective space
arXiv: Complex Variables, 2014Co-Authors: Risto Korhonen, Kazuya TohgeAbstract:Tropical Nevanlinna theory, introduced by Halburd and Southall as a tool to analyze integrability of ultra-discrete equations, studies the growth and complexity of continuous piecewise linear real functions. The purpose of this paper is to extend tropical Nevanlinna theory to n-dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves, and by proving a tropical analogue of Cartan's second Main Theorem. It is also shown that in the 1-dimensional case this result implies a known tropical second Main Theorem due to Laine and Tohge.
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tropical nevanlinna theory and second Main Theorem
Proceedings of The London Mathematical Society, 2011Co-Authors: Ilpo Laine, Kazuya TohgeAbstract:We present a version of the tropical Nevanlinna theory for real-valued, continuous, piecewise linear functions on the real line. In particular, a tropical version of the second Main Theorem is proved. Applications to some ultra-discrete equations are given.
