Lecture Notes on Diophantine Analysis
Lecture Notes on Diophantine Analysis
复制标题
丢番图分析讲义
DOI：
10.1007/978-88-7642-517-2
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
U. Zannier
中科院分区：
文献类型：
--
作者：
U. Zannier
1.经典丢番图方程：线性与二次方程，佩尔方程，丢番图逼近，同余。关于Pell方程及exp（n）和pi的无理性的补充笔记。- 2.丢番图方程和有理逼近的图厄定理：策略描述和详细证明。后来的改进。关于曲线上整点和龙格定理的补充。笔记。- 3.数域上的高度和丢番图方程：乘积公式，韦尔和马勒高度，数域上的丢番图近似，S-单位方程及其应用。关于函数域中abc定理和代数函数及其值的乘法依赖性的补充。笔记。- 4. G_m^n的子簇上的高：平面曲线上的挠点和G_m^n的子簇上的小高代数点。代数子群的结构。张必鲁定理及其在S-单位方程中的应用关于R^n的离散和闭子群以及Skolem-Mahler-Lech定理的补充。笔记。- 5. S单位方程一个尖锐的定量S-单位定理，显式Pade'逼近和大解的计数，小解的计数。数量S单位定理的应用。笔记。- F.的附录。Amoroso：Bounds for the height：Generalized Lehmer problem，Dobrowolski lower bounds.簇的高度和下界向高维的扩展使张量定理变得尖锐。
1. Classical Diophantine Equations: linear and quadratic equations, Pell Equation, Diophantine Approximation, congruences. Supplements on Pell equations and irrationality of exp(n) and pi. Notes.- 2. Thue's theorems on Diophantine Equations and rational approximations: Description of strategy and detailed proofs. Later refinements. Supplements on integral points on curves and Runge's theorem. Notes.- 3. Heights and Diophantine equations over number fields: Product formulas, Weil and Mahler heights, Diophantine approximation in number fields, the S-unit equation and its applications. Supplements on the abc-theorem in function fields and on multiplicative dependence of algebraic functions and their values. Notes.- 4. Heights on subvarieties of G_m^n: Torsion points on plane curves and algebraic points of small height on subvarieties of G_m^n. Structure of algebraic subgroups. Theorems of Zhang and Bilu and applications to the S-unit equation. Supplements on discrete and closed subgroups of R^n and on the Skolem-Mahler-Lech theorem. Notes.- 5. The S-unit equation. A sharp quantitative S-unit theorem explicit Pade' approximations and the counting of large solutions counting of small solutions. Applications of the quantitative S-unit theorem. Notes.- Appendix by F. Amoroso: Bounds for the height: Generalized Lehmer problem, Dobrowolski lower bounds. Heights of varieties and extensions of lower bounds to higher dimensions sharp quantitative Zhang's theorem.