A complete diophantine characterization of the rational torsion of an elliptic curve

We give a complete characterization for the rational torsion of an elliptic curve in terms of the (non-)existence of integral solutions of a system of diophantine equations.     

The Number of Solutions of Polynomial-Exponential Equations

We will give explicit bounds for the numberof solutions of polynomial-exponential equations.In contrast to earlier work, the bounds areindependent of the coefficients of the equations,and they are of only single exponential growthin the number of coefficients.     

关于直径为4的整树

给出了直径为4的整树的一个实用的充要条件,讨论了几个基本的公开问题, 并给出了直径为4的整树的一些新类.     

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.

     

一个指数Diophantine方程的正整数解

本文研究了一个纯指数Diophantine方程的求解问题,利用有关Lucas数本原素因数的存在性方面的新近结果,获得了该方程的全部正整数解.     

On the Frobenius problem for Beatty sequences

We study the solvability of linear diophantine equations in two variables within the context of Beatty sequences.     

Mumford's Degree of Contact and Diophantine Approximations

The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show that destabilizing flags of Chow-unstable projective varieties provide systems of diophantine approximations which are better than those given by Schmidt's subspace theorem, and we give examples of these systems.     

关于广义Ramanujan-Nagell方程的一个猜想

本文研究广义Ramanujan-Nagell方程的正整数解.利用初等数论方法,证实了杨仕椿关于广义Ramanujan-Nagell方程的一个猜想.     

A General Algorithm for the MacMahon Omega Operator

In his famous book Combinatory Analysis MacMahon introduced Partition Analysis (Omega Calculus) as a computational method for solving problems in connection with linear diophantine inequalities and equations. The technique has recently been given a new life by G.E. Andrews and his coauthors, who had the idea of marrying it with the tools of to-days Computer Algebra.The theory consists of evaluating a certain type of rational function of the form A()^-1 B(1/)^-1 by the MacMahon operator. So far, the case where the two polynomials A and B are factorized as products of polynomials with two terms has been studied in details. In this paper we study the case of arbitrary polynomials A and B. We obtain an algorithm for evaluating the operator using the coefficients of those polynomials without knowing their roots. Since the program efficiency is a persisting problem in several-variable polynomial Calculus, we did our best to make the algorithm as fast as possible. As an application, we derive new combinatorial identities.AMS Subject Classification: 05A17, 05A19, 05E05, 15A15, 68W30.     

A Fourth Derivative Test for Exponential Sums

We give an upper bound for the exponential sum ^M _m=1 e^2if(m) in terms of M and , where is a small positive number which denotes the size of the fourth derivative of the real valued function f. The classical van der Corput's exponent 1/14 is improved into 1/13 by reducing the problem to a mean square value theorem for triple exponential sums.