arXiv subjects

Julie Tzu-Yueh Wang

Publications and source records attributed to Julie Tzu-Yueh Wang.

At least 19 recordsLinked to original sources

arXiv2026

Families of Unit Equations and Exponential Diophantine Problems via Integral Points

This paper investigates the distribution of integral points on projective varieties via two distinct methods: the Ru-Vojta theorem and our higher-dimensional generalization of the Huang-Levin-Xiao inequalities. These approaches operate under distinct geometric conditions, specifically the transverse and proper intersections of boundary divisors. Applying this framework, we prove degeneracy results for the solution sets of two classes of one-parameter families of unit equations, differentiated by the degrees of their polynomial coefficients. Finally, we extend previous greatest common divisor (GCD) estimates to derive new results for specific exponential Diophantine equations and the distribution of digits in $q$-adic representations.

math.NT↗

arXiv2026

Hyperbolicity and GCD for n+1 divisors with non-empty intersection

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local conditions on beta constants or intersection multiplicities, we prove that all entire curves are algebraically degenerate. Our approach extends the method of Levin-Huang-Xiao to higher dimensions, establishing a second main theorem for regular sequences of closed subschemes. This also yields a GCD-type estimate in the same geometric setting.

math.CV↗

arXiv2026

Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case $n=2$ (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.

math.CV↗

arXiv2025

Campana conjecture for coverings of toric varieties over function fields

We first prove Vojta's abc conjecture over function fields for Campana points on projective toric varieties with high multiplicity along the boundary. As a consequence, we obtain a version of Campana's conjecture on finite coverings of projective toric varieties over function fields.

math.AG↗

arXiv2025

Campana's orbifold conjecture for numerically equivalent divisors

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$ and $D:=D_1+\cdots+D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $Δ=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i$ and let $f:\mathbb C\to (X,Δ) $ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $ (X,Δ_{\ell}) $ is of general type, where $Δ_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate.

math.CV↗

arXiv2024

Defect relation of $n+1$ components through the GCD method]

This paper studies the defect relation through the GCD method. In particular, among other results, we extend the defect relation result of Chen, Huynh, Sun and Xie to moving targets. The truncated defect relation is also studied. Furthermore, we obtain the degeneracy locus, which can be determined effectively and is independent of the maps under the consideration.

math.CV↗

arXiv2023

Vojta's abc Conjecture for algebraic tori and applications over function fields

We prove Vojta's generalized abc conjecture for algebraic tori over function fields with exceptional sets that can be determined effectively. Additionally, we establish a version of the conjecture for toric varieties. As an application, we investigate the Lang-Vojta Conjecture for varieties of log general type that are ramified covers of $\mathbb G_m^n$ over function fields. In particular, we consider the case of $ \mathbb P^n\setminus D$, where $D$ is an algebraic curve over a function field in $\mathbb P^n$ with $n+1$ irreducible components and $°D\ge n+2$. Our methods also apply to the complex situation, enabling us to find explicit exceptional sets for the corresponding case of Vojta's general abc conjecture (complex version) and the Green-Griffith-Lang conjecture.

math.NT↗

arXiv2023

Simply connectedness and hyperbolicity

We generalize to arbitrary dimension our previous construction of simply connected weakly-special but not special varieties. We show that they satisfy the function field and complex analytic part of Campana's conjecture. Moreover, we give the first examples, in any dimension, of smooth simply connected nonisotrivial projective varieties of general type that satisfy the function field Lang's conjecture.

math.AG↗

arXiv2023

A complex case of Vojta's general abc conjecture and cases of Campana's orbifold conjecture

We proved a truncated second main theorem of level one with explicit exceptional sets for analytic maps into $\mathbb P^2$ intersecting the coordinate lines with sufficiently high multiplicities. As applications, we studied some cases of Campana's orbifold conjecture for $\mathbb P^2$ and finite ramified covers of $\mathbb P^2$ with three components admitting sufficiently large multiplicities.

math.CV↗

arXiv2021

Greatest common divisors with moving targets and consequences for linear recurrence sequences

We establish consequences of the moving form of Schmidt's Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving $S$-unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin and others.

math.NT↗

arXiv2021

Divisibility of polynomials and degeneracy of integral points

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.

math.NT↗

arXiv2021

A truncated second main theorem for algebraic tori with moving targets and applications

We establish a second main theorem for algebraic tori with slow growth moving targets with truncation to level 1. As the first application of this result, we prove the Green-Griffith-Lang conjecture for projective spaces with $n+1$ components in the context of moving targets of slow growth. Then we discuss the integrability of the ring of exponential polynomials in the ring of entire functions as another application.

math.CV↗

arXiv2021

The Ru-Vojta result for subvarieties

In their recent article, Min Ru and Paul Vojta, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt's subspace theorem. In this note, we extend their result by replacing the divisors by closed subschemes.

math.NT↗

arXiv2021

Nonspecial varieties and Generalized Lang-Vojta conjectures

We construct a family of fibered threefolds $X_m \to (S , Δ)$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,Δ)$ of general type. Following Campana, the threefolds $X_m$ are called \emph{weakly special} but not \emph{special}. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if $m$ is big enough the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalizations of known conjectures on exceptional loci that fit into Campana's program and prove some cases over function fields.

math.AG↗

arXiv2020

On Pisot's $d$-th root conjecture for function fields and related GCD estimates

We propose a function-field analog of Pisot's $d$-th root conjecture on linear recurrences, and prove it under some "non-triviality" assumption. Besides a recent result of Pasten-Wang on B{ü}chi's $d$-th power problem, our main tool, which is also developed in this paper, is a function-field analog of an GCD estimate in a recent work of Levin and Levin-Wang. As an easy corollary of such GCD estimate, we also obtain an asymptotic result.

math.NT↗

arXiv2019

Greatest common divisors of integral points of numerically equivalent divisors

We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substituted by a more general version of Schmidt's subspace theorem of Ru--Vojta in our proof.

math.NT↗

arXiv2019

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.

math.CV↗

arXiv2015

On non-Archimedean curves omitting few components and their arithmetic analogues

Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When X is a rational ruled surface and D_1 and D_2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into $X\setminus D_1 \cup D_2$. Using a dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over the integers or the ring of integers of an imaginary quadratic field.

math.CV↗