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Si Duc Quang

Publications and source records attributed to Si Duc Quang.

At least 37 records · Page 2Linked to original sources
arXiv2023

Truncated second main theorem for non-Archimedean meromorphic maps

Let $\mathbb F$ be an algebraically closed field of characteristic $p\ge 0$, which is complete with respect to a non-Archimedean absolute value. Let $V$ be a projective subvariety of $\mathbb P^M(\mathbb F)$. In this paper, we will prove some second main theorems for non-Archimedean meromorphic maps of $\mathbb F^m$ into $V$ intersecting a family of hypersurfaces in $N-$subgeneral position with truncated counting functions.

math.AG↗
arXiv2022

Meromorphic functions on annuli sharing finite sets with truncated multiplicity

The purpose of this paper has twofold. The first is to establish a second main theorem for meromorphic functions on annuli and meromorphic function targets (may not be small functions) with truncated counting functions (truncation level 1) and with a detailed estimate for the error term. The second is to show that if the polynomial $$P_S(w)=(w-a_1)\cdots (w-a_q)$$ is a uniqueness polynomial for admissible meromorphic functions on an annulus $\mathbb A(R_0)$ such that $P'_S(w)$ has exactly $k$ distinct zeros and $q>\frac{(5k+7)\ell}{2\ell-175}$, then the set $S=\{a_1,\ldots,a_q\}$ is a finite range set with truncation level $\ell$ for admissible meromorphic functions on $\mathbb A(R_0)$. This result extends the previous result on the finite range set (with truncation level $\ell=\infty$) for holomorphic functions on $\mathbb C$ of H. Fujimoto.

math.CV↗
arXiv2022

Holomorphic maps from complex discs intersecting families of hypersurfaces

In this paper, we establish second main theorems for holomorphic maps with finite growth index on complex discs intersecting families of hypersurfaces (moving and fixed) in projective varieties, where the small term is detailed estimate for various cases. Our results are generalizations and extensions of many previous second main theorems for holomorphic maps from $\mathbb C$ intersecting with hypersurfaces (moving and fixed) or moving hyperplanes.

math.CV↗
arXiv2021

Degeneracy theorems for meromorphic mappings of a complete Kähler manifold sharing hyperplanes in a projective space

Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0 C+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k\ (2\le k\le n+1)$ meromorphic mappings sharing hyperplanes is also given. Our result generalize the previous result in the case where the mappings from $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$.

math.CV↗
arXiv2021

Meromorphic mappings into projective varieties with arbitrary families of moving hypersurfaces

In this paper, we prove a general second main theorem for meromorphic mappings into a subvariety $V$ of $\mathbb P^N(\mathbb C)$ with an arbitrary family of moving hypersurfaces. Our second main theorem generalizes and improves all previous results for meromorphic mappings with moving hypersurfaces, in particular for meromorphic mappings and families of moving hypersurfaces in subgeneral position. The method of our proof is different from that of previous authors used for the case of moving hypersurfaces.

math.CV↗
arXiv2021

Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds

Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if $k$ differentibility nondegenerate meromorphic mappings $f^1,\ldots,f^k$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_ρ)$ and sharing few hyperplanes in subgeneral position regardless of multiplicity then $f^1\wedge\cdots\wedge f^k\equiv 0$. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of $M$ into $\mathbb P^n(\mathbb C)$ sharing $q\ (q\sim 2N-n+3+O(ρ))$ hyperplanes in $N-$subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of $\mathbb C^m$ and extend some previous results for the case of mappings on Kähler manifold.

math.CV↗
arXiv2021

Generalizations of degeneracy second main theorem and Schmidt's subspace theorem

In this paper, by introducing the notion of "\textit{distributive constant}" of a family of hypersurfaces with respect to a projective variety, we prove a second main theorem in Nevanlinna theory for meromorphic mappings with arbitrary families of hypersurfaces in projective varieties. Our second main theorem generalizes and improves previous results for meromorphic mappings with hypersurfaces, in particular for algebraically degenerate mappings and for the families of hypersurfaces in subgeneral position. The analogous results for the holomorphic curves with finite growth index from a complex disc into a project variety, and for meromorphic mappings on a complete Kähler manifold are also given. For the last aim, we will prove a Schmidt's subspace theorem for an arbitrary families of homogeneous polynomials, which is the counterpart in Number theory of our second main theorem. Our these results are generalizations and improvements of all previous results.

math.CV↗
arXiv2020

Non-integrated defect relation for meromorphic mappings from a Kähler manifold with hypersurfaces of a projective variety in subgeneral position

In this paper, we establish a truncated non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting a family of hypersurfaces located in subgeneral position, where the truncation level of the defect is explicitly estimated. Our result generalizes and improves previous ones. In particular, when the family of hypersurfaces located in general position, our theorem will implies the previous result of Min Ru-Sogome. In the last part of this paper we will apply ours to study the distribution of the Gauss map of minimal surfaces.

math.CV↗
arXiv2020

Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes

Let $M$ be a complete Kähler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0 2n+1+α+ρK)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $α<1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level $n+1$ then $f^1=f^2=f^3.$ Our results generalize the finiteness and uniqueness theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplanes in general position with truncated multiplicity.

math.CV↗
arXiv2018

Generalization of uniqueness theorem for meromorphic mappings sharing hyperplanes

In this article, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into $¶^n(\C )$ which share few hyperplanes regardless of multiplicities. Our result improves the previous result in this topic. Moreover our proof is simpler than the previous proofs.

math.CV↗
arXiv2018

Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

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.

math.NT↗
arXiv2017

Degeneracy second main theorems for meromorphic mappings into projective varieties with hypersurfaces

The purpose of this paper has twofold. The first is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. 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. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case.

math.CV↗
arXiv2017

Degeneracy and finiteness theorems for meromorphic mappings in several complex variables

In this article, we prove that there are at most two meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 2)$ sharing $2n+2$ hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. We also show that if three meromorphic mappings $f^1,f^2,f^3$ of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)\ (n\geqslant 5)$ share $2n+1$ hyperplanes in general position with truncated multiplicity then the map $f^1\times f^2\times f^3$ is linearly degenerate.

math.CV↗
arXiv2016

Non-integrated defect relation for meromorphic maps from a Kähler manifold intersecting hypersurfaces in subgeneral of $\mathbb P^n(\mathbb C)$

In this article, we establish a truncated non-integrated defect relation for meromorphic mappings from an $m$-dimensional complete Kähler manifold into $\mathbb P^n(\mathbb C)$ intersecting $q$ hypersurfaces $Q_1,...,Q_q$ in $k$-subgeneral position of degree $d_i$, i.e., the intersection of any $k+1$ hypersurfaces is emptyset. We will prove that $$ \sum_{i=1}^qδ_f^{[u-1]}(Q_i)\le (k-n+1)(n+1)+ε+\frac{ρu(u-1)}{d}, $$ where $u$ is explicitly estimated and $d$ is the least common multiple of $d_i'$s. Our result generalizes and improves previous results. In the last part of this paper we will apply this result to study the distribution of the Gauss map of minimal surfaces.

math.CV↗