Jones在Hilbert第十问题及其相关课题上的工作：为Jones教授访华而作

本文是有关Hilbert第十问题近代成果(特别是James P.Jones教授的工作)的综述报告。它由六个部分组成:1.Hilbert第十问题;2.9未知数定理;3.通用不定方程;4.不定方程前缀量词分类;5.Diophantine表示;6.Hilbert第十问题的应用。文中还提到了作者的一些新结果,例如~(11)over Z不可判定。     

一类拟齐次多项式中心的极限环个数

考虑一类具有全局中心的(m,1)型拟齐次多项式平面微分系统,通过探讨阿贝尔积分的零点个数,分别研究该系统在n次多项式和在(n,1)型拟齐次多项式扰动下,从中心的周期环域分支出来的极限环个数,给出了这些个数的上界并证明它们是可达的.     

一类五次扰动Hamiltonian系统的Abel积分零点个数

研究一类五次扰动Hamiltonian系统的Abel积分零点个数上界.证明所研究的Abel积分的生成元构成精度为1的Chebeyshev系统,得到Abel积分零点个数上界是4(考虑零点重数).并指出前人文献中关于Abel积分零点个数上界的研究存在的错误,给出了最新结果.     

函数分解问题的判定性研究

函数分解问题是一个重要的数学问题,它被应用于多变量公钥密码系统的设计中.作者主要关注函数分解问题的判定性问题,将整系数多变元多项式方程求解问题与函数分解问题建立起了一定的联系,从而利用著名的希尔伯特第十问题的不可判定性来对函数分解问题的判定性问题进行研究.     

积分问题中分段估值的解题方法

细致分析和研究积分不等式、积分值序列的极限、含参变量积分的极限等若干方面诸多问题的解决过程,提炼出其中蕴含的一类解题方法--分段估值法.     

一类三次系统的中心判定问题

对于一类具有三次衄线解x~2(x-1)-y~2-1=0,通过点(1,0)的直线解和中心-焦点型奇点的三次系统,证明了它以原点为中心的充要条件是它的前五阶焦点量全为零.这些中心条件是通过构造积分因子得以验证的.     

一类哈密顿系统的Abel积分之比单调性的判据

研究了一类哈密顿系统的两个Abel积分比值的单调性的条件,指出这个单调性条件可由文中给出的两个判定函数直接确定.     

一类特殊二次微分系统的分支问题

本文考虑［１，§１４］中提到的二次系统的极限环数目和分支问题，分别在δ１轴和（δ１，δ２）平面上画出相应的分支图．     

一类Hamilton系统的Abel积分比值单调性的判别条件

用直接计算的方法对一类Hamilton系统的两个Abel积分比值的单调性进行讨论,指出该单词性条件可由两个判定函数直接确定．     

A cubic system with thirteen limit cycles

We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

An elementary and constructive solution to Hilbert's 17th Problem for matrices

We give a short and elementary proof of a theorem of Procesi, Schacher and (independently) Gondard, Ribenboim that generalizes a famous result of Artin. Let be an symmetric matrix with entries in the polynomial ring . The result is that if is positive semidefinite for all substitutions , then can be expressed as a sum of squares of symmetric matrices with entries in . Moreover, our proof is constructive and gives explicit representations modulo the scalar case.

     

Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of

We give the first examples of infinite sets of primes such that Hilbert's Tenth Problem over has a negative answer. In fact, we can take to be a density 1 set of primes. We show also that for some such there is a punctured elliptic curve over such that the topological closure of in has infinitely many connected components.

     

Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers

Using Poonen's version of the ``weak vertical method' we produce new examples of ``large' and ``small' rings of algebraic numbers (including rings of integers) where and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur's conjecture on the topology of rational points does not hold.

     

Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

The number and distributions of limit cycles for a class of cubic near-Hamiltonian systems

This paper concerns with the number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist 9-11 limit cycles is proved. The different distributions of limit cycles are given by using methods of bifurcation theory and qualitative analysis, among which two distributions of eleven limit cycles are new.     

Abel-like differential equations with no periodic solutions

We present various criteria for the non-existence of positive periodic solutions of generalized Abel differential equations with periodic coefficients that can change sign. As an application, we obtain some families of planar vector fields without limit cycles.