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Si Duc Quang

Publications and source records attributed to Si Duc Quang.

At least 19 recordsLinked to original sources
arXiv2026

A new degenerated second main theorem for meromorphic mappings with hypersurfaces

We establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into a projective variety and a family of hypersurfaces in subgeneral position. The above bound of the total defect obtained from our result is better than that of the previous results. Moreover, in our result, the truncation level of the counting functions is estimated explicitly and independently of the number of hypersurfaces. Especially, we do not need the assumption that the family of hypersurfaces must satisfy the Bezout properties as imposed in some earlier studies, but only a weak Bezout property.

math.AG↗
arXiv2026

Meromorphic functions bi-weighted weakly sharing pairs of small functions

Two meromorphic functions $f$ and $g$ are said to weakly share a small function $a$ with bi-weight $(n,k)$ if the functions $f-a$ and $g-a$ have the same zeros with multiplicities truncated at level $n+1$, while zeros whose multiplicities exceed $k$ are disregarded. In this article, we show that if $f$ and $g$ weakly share three distinct small functions with suitable bi-weights and are not related by a quasi-Möbius transformation, then for every other small function $c$, the counting function $N(r,ν_f^c)$ is asymptotically equivalent to the characteristic function $T(r,f)$. Moreover, the truncated counting function $N_{(3}(r,ν_f^c)$, which counts only zeros of multiplicity at least $3$, is negligible. As an application, we further prove that $f$ and $g$ must be related by a quasi-Möbius transformation provided that they satisfy an additional condition, which is weaker than the usual assumption that they share a fourth pair of small functions.

math.CV↗
arXiv2026

Characteristic functions of two meromorphic functions weakly sharing three small functions with bi-weights

Two meromorphic functions $f$ and $g$ are said to weakly share a small function $a$ with bi-weight $(n,k)$ if the functions $f-a$ and $g-a$ have the same zeros with multiplicities truncated at level $n+1$, while zeros whose multiplicities exceed $k$ are disregarded. In this article, we show that if two meromorphic functions $f$ and $g$ weakly share three small functions $a_i\ (1\le i\le 3)$ with bi-weights $(n_i,k)$ satisfying $n_1n_2n_3>n_1+n_2+n_3+2$ then $$(1-ε-δ_ε)T(r, f)\le (2+ε+δ_ε)T(r, g)+S(r, g)$$ for every positive number $ε$, where $δ_ε$ is explicitly estimated depending only on $ε$ and $k$, so that $δ_ε$ tends to zero as $k$ tends to $+\infty$.

math.CV↗
arXiv2026

Total curvature of a complete minimal surface and the modified defect relation of a Fermat hypersurface for the Gauss map

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset\mathbb R^m$ into $\mathbb P^n(\mathbb C)\ (n=m-1)$ with only a single Fermat hypersurface $Q$ of $\mathbb P^n(\mathbb C)$. In particular, we show that $S$ must have finite total curvature if the image $g(S)$ intersects $Q$ with only a finite number of times and the degree of $Q$ is sufficiently large.

math.DG↗
arXiv2026

Meromorphic mappings into projective varieties intersecting arbitrary families of moving hypersurfaces

In this paper, we establish a general second main theorem for meromorphic mappings from $\mathbb C^m$ into a subvariety $V$ of $\mathbb P^n(\mathbb C)$ with respect to an arbitrary family of slowly moving hypersurfaces $\mathcal Q=\{Q_1,\ldots,Q_q\}$. In contrast to the usual setting, the mapping is not required to be algebraically nondegenerate over the field $\mathcal K_{\mathcal Q}$. Moreover, the truncation levels of the counting functions are explicitly estimated, and the total defect bound is given by $Δ_{\mathcal Q,V}(3\dim V-1)$, which is independent of the mapping $f$, where $Δ_{\mathcal Q,V}$ denotes the distributive constant of $\mathcal Q$ with respect to $V$.

math.CV↗
arXiv2026

A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions

By using Brownian motion and stochastic calculus, we establish a second main theorem for holomorphic curves into a projective subvariety $V\subset\mathbb P^n(\mathbb C)$ with an arbitrary family $\mathcal Q$ of $q$ hypersurfaces $Q_1,\ldots,Q_q$ concerning its distributive constant $Δ_{\mathcal Q,V}$. In our result, the counting functions are truncated to level $H_V(d)-1$, where $d=lcd(°Q_1,\ldots,°Q_d)$ and $H_V(d)$ is the Hilbert function of $V$. As an application of the second main theorem, we give a uniqueness theorem for holomorphic curves from $\mathbb C$ into $V$ sharing an arbitrary family of hypersurfaces regardless of multiplicity.

math.CV↗
arXiv2026

Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. The theorems are obtained in several settings, including holomorphic curves on $\mathbb{C}$, annuli, complex discs with finite growth index, and Kähler manifolds. We obtain a total defect bound that improves upon the previously known results. As an application, we establish a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

math.CV↗
arXiv2024

Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in subgeneral position

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset\mathbb R^m$ into a $k$-dimension projective subvariety $V\subset\mathbb P^n(\mathbb C)\ (n=m-1)$ with hypersurfaces $Q_1,\ldots,Q_q$ of $\mathbb P^n(\mathbb C)$ in $N$-subgeneral position with respect to $V\ (N\ge k)$. In particular, we give the upper bound for the number $q$ if the image $g(S)$ intersects each hypersurfaces $Q_1,\ldots,Q_q$ a finite number of times and $g$ is nondegenerate over $I_d(V)$, where $d=lcm(°Q_1,\ldots,°Q_q)$, i.e., the image of $g$ is not contained in any hypersurface $Q$ of degree $d$ with $V\not\subset Q$. Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.

math.DG↗
arXiv2024

Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position

Let $A$ be an annular end of a complete minimal surface $S$ in $\mathbb R^m$ and let $V$ be a $k$-dimension projective subvariety of $\mathbb P^n(\mathbb C)\ (n=m-1)$. Let $g$ be the generalized Gauss map of $S$ into $V\subset\mathbb P^n(\mathbb C)$. In this paper, we establish a modified defect relation of $g$ on the annular end $A$ for $q$ hypersurfaces $\{Q_i\}_{i=1}^q$ of $\mathbb P^n(\mathbb C)$ in $N$-subgeneral position with respect to $V$. Our result implies that the image $g(A)$ cannot omit all $q$ hypersurfaces $Q_1,\ldots,Q_q$ if $g$ is nondegenerate over $I_d(V)$ and $q>\frac{(2N-k+1)(M+1)(M+2d)}{2d(k+1)}$, where $M=H_V(d)-1$ and $d$ is the least of common multiple of $°Q_1,\ldots,°Q_q$. As our best knowledge, it is the first time the value distribution of the Gauss map on an annular end of a minimal surfaces with hypersurface targets is studied, in particular the product into sum inequality for holomorphic curves on Riemann surfaces with hypersurfaces targets is presented. This our result has been used to study the unicity of the gauss maps in the recent work of C. Lu and X. Chen [14].

math.AG↗
arXiv2024

General form of second main theorem on generalized $p$-Parabolic manifolds for arbitrary closed subschemes

By introducing the notion of distributive constant for a family of closed subschemes, we establish a general form of the second main theorem for algebraic nondegenerate meromorphic mappings from a generalized $p$-Parabolic manifold into a projective variety with arbitrary families of closed subschemes. As its consequence, we give a second main theorem for such meromorphic mappings intersecting arbitrary hypersurfaces with an explicitly truncation level for the counting functions.

math.CV↗
arXiv2024

Meromorphic mappings on Kähler manifolds weakly sharing hyperplanes in $\mathbb P^n(\mathbb C)$

In this paper, we study the uniqueness problem for linearly nondegenerate meromorphic mappings from a Kähler manifold into $\mathbb P^n(\mathbb C)$ satisfying a condition $(C_ρ)$ and sharing hyperplanes in general position, where the condition that two meromorphic mappings $f,g$ have the same inverse image for some hyperplanes $H$ is replaced by a weaker one that $f^{-1}(H)\subset g^{-1}(H)$. Moreover, we also give some improvements on the uniqueness problem and algebraic dependence problem of meromorphic mappings which share hyperplanes and satisfy $(C_ρ)$ conditions for different non-negative numbers $ρ$.

math.CV↗
arXiv2024

Defect relation for holomorphic maps from complex discs into projective varieties and hypersurfaces

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also is generalizes and improves many previous second main theorems for holomorphic maps from $\mathbb C$ intersecting hypersurfaces (moving and fixed) in projective varieties.

math.CV↗
arXiv2024

Improvement of non-integrated defect relation for meromorphic maps from Kähler manifolds

The purpose of this paper is to establish a non-integrated defect relation for meromorphic mappings from a complete Kähler manifold into a projective variety intersecting an arbitrary family of hypersurfaces with explicit truncation level. In our result, both the total defect and the truncation level are estimated independently of the number of involving hypersurfaces. Our result generalizes and improves the previous results in this topic.

math.CV↗
arXiv2024

Second main theorem and uniqueness problem of meromorphic functions with finite growth index sharing five small functions on a complex disc

This paper has twofold. The first is to establish a second main theorem for meromorphic functions on the complex disc $Δ(R_0)\subset\mathbb C$ with finite growth index and small functions, where the counting functions are truncated to level $1$ and the small term is more detailed estimated. The second is to prove a generalization and improvement of the five values theorem of Nevanlinna for the case of five small functions on the complex disc $Δ(R_0)$.

math.CV↗
arXiv2024

On generalized Gauss maps of minimal surfaces sharing hypersurfaces in a projective variety

In this article, we study the uniqueness problem for the generalized gauss maps of minimal surfaces (with the same base) immersed in $\mathbb R^{n+1}$ which have the same inverse image of some hypersurfaces in a projective subvariety $V\subset\mathbb P^n(\mathbb C)$. As we know, this is the first time the unicity of generalized gauss maps on minimal surfaces sharing hypersurfaces in a projective varieties is studied. Our results generalize and improve the previous results in this field.

math.DG↗
arXiv2023

Some generalizations of Schmidt's subspace theorem

The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary families of closed subschemes in algebraic projective varieties.

math.NT↗