On the Zariski-Density of Integral Points on a Complement of Hyperplanes in P^n

The Dimensions of Integral Points and Holomorphic Curves on the Complements of Hyperplanes

In this article we completely determine the possible dimensions of integral points and holomorphic curves on the complement of a union of hyperplanes in projective space. Our main theorems generalize a result of Evertse and Gyory, who determined when all sets of integral points (over all number fields) on the complement of a union of hyperplanes are finite, and a result of Ru, who determined when all holomorphic maps to the complement of a union of hyperplanes are constant. The main tools used are the S-unit lemma and its analytic analogue, Borel's lemma.

关于射影空间中某些子簇余集上整点的潜在密度的研究

On potential density of integral points on the complement of some subvarieties in the projective space

本文研究了数域上 $n$ 维射影空间中特定闭子簇余集上整点的潜在密度，推广了 Corvaja-Zucconi 在低维射影空间中的相关结果。

有限域 $\mathbf{F}_{q^r}$ 中避开仿射超平面的本原元，对于 $q=4$ 和 $q=5$

Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$

本文证明了当 $q=4$ 和 $q=5$ 且 $r$ 足够大时，有限域 $\mathbf{F}_{q^r}$ 中存在避开 $r$ 个仿射超平面的本原元，填补了 $q\ge 7$ 与小 $q$ 情形之间的研究空白。

The Hilbert Property for integral points of affine smooth cubic surfaces

In this paper we prove that the set of $S$-integral points of the smooth cubic surfaces in $\mathbb{A}^3$ over a number field $k$ is not thin, for suitable $k$ and $S$. As a corollary, we obtain results on the complement in $\mathbb{P}^2$ of a smooth cubic curve, improving on Beukers' proof that the $S$-integral points are Zariski dense, for suitable $S$ and $k$. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface $x^3+y^3+z^3=1$ form a non-thin…

Integral Points and Relative Sizes of Coordinates of Orbits in P^N

We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on P^N so that (S,D)-integral points in each orbit are Zariski-non-dense. This condition is geometric, and for dimension 1 it corresponds precisely to Silverman's hypothesis that the second iterate of the map is not a polynomial. In fact, we will prove a more precise formulation comparing local heights outside S to the global height. For hyperplanes, this amounts to comparing logarithmic sizes of the coordinate…

某些仿射三次曲面的Brauer群

Brauer groups of certain affine cubic surfaces

该研究针对特征0域上光滑三次曲面奇异超平面截面的补仿射曲面，确定了代数闭包上其Brauer群的Galois模结构，给出三线并截面对应的2、3阶超越元有理数域实例，并应用于整Hasse原理的整Brauer-Manin障碍研究。