A simple proof of Schmidt–Summerer’s inequality

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W.M. Schmidt and L. Summerer.     

On the number of Diophantine m-tuples in finite fields

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine m-tuples in finite fields, which is due to A. Dujella and M. Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).     

On the number of Diophantine <Emphasis Type="Italic">m</Emphasis>-tuples

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m=2, 3 and 4. In this paper, we derive asymptotic estimates for the number of Diophantine pairs, triples and quadruples with elements less than given positive integer N. The author was supported by the Ministry of Science and Technology, Republic of Croatia, grants 0037110 and 037-0372781-2821.     

A new technique for obtaining diophantine representations via elimination of bounded universal quantifiers

M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 83–92. Original article Translated by Yu. V. Matiyasevich. The research described in this work was made possible in part by Grant No. 94-01-01030 from the Russsian Foundation for Fundamental Research and Grant No. R43000 from the International Science Foundation.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

关于不定方程x~2-Dy~2=c的最小正整数解

给出一般二元二次不定方程最小正整数解的一个判定准则 ,确定了几类二元二次不定方程和Pell方程的最小正整数解 ,推广了 [1 ]、[2 ]中的两个结果     

A characterisation of all feasible solutions to an integer program

It is shown how the dual of Fourier–Motzkin elimination can be applied to eliminating the constraints of an Integer Linear Program. The result will, in general, be to reduce the Integer Program to a single Diophantine equation together with a series of Linear homogeneous congruences. Extreme continuous solutions to the Diophantine equation give extreme solutions to the Linear Programming relaxation. Integral solutions to the Diophantine equation which also satisfy the congruences give all the solutions to the Integer Program.     

A diophantine definition of integers in the rings of rational numbers

The author considers rings of rational numbers which are integral at all the primes except, possibly, primes contained in a finite set. In such rings a Diophantine definition of ? is constructed to show that all the recursively enumerable subsets of the ring are Diophantine.     

Joint Continuity of Lyapunov Exponent for Finitely Smooth Quasi-periodic Schr?dinger Cocycles

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schr?dinger cocycles with the C~2 cos-type potentials. In particular, the Lyapunov exponent is log-H?lder continuous at each Diophantine frequency.     

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。     

An efficient algorithm for solving rank one perturbed linear Diophantine systems using Rosser��s approach

Recently, we described a generalization of Rosser’s algorithm for a single linear Diophantine equation to an algorithm for solving systems of linear Diophantine equations. Here, we make use of the new formulation to present a new algorithm for solving rank one perturbed linear Diophantine systems, based on using Rosser’s approach. Finally, we compare the efficiency and effectiveness of our proposed algorithm with the algorithm proposed by Amini and Mahdavi-Amiri (Optim Methods Softw 21:819–831, 2006).     

On a Family of Diophantine Difference Equations

The family of Diophantine difference equations considered byJ. H. Conway, H. T. Croft, P. Erds and M. J. T. Guy can be solvedin closed form when the parameter is a prime. The associatedcharacterisation problem is shown to be related to the determinationof certain constants, which also happen to be coefficients inthe asymptotic formula for the associated exponential sum.     

On the Diophantine Equation x2 - kxy ＋ y2 ＋ 1x = 0

Let l be a given nonzero integer.The authors give an explicit characterization of the positive integer k that makes the Diophantine equation x2-kxy+y2+lx=0 have infinitely many positive integer solutions(x,y).     

Euclid域中Diophantus方程的整数解

本文研究了两个典型Diophantus方程的整数解的问题.利用二次域中的重要理论和二次代数整数环中算术基本定理,获得了两个典型Diophantus方程在Euclid域中的所有整数解,推广了文献[6]的结果.     

An inhomogeneous Jarník theorem

We compute the generalized Hausdorff measure of sets of points in R ^s which satisfy an inhomogeneous system of Diophantine inequalities infinitely often. This provides an inhomogeneous analogue of a classical result of Jarník on simultaneous Diophantine approximation.     

关于不定方程组x-1=3py^2，x^2＋x＋1=3z^2

设P为素数，利用同余及高次丢番图方程的一些结果证明了不定方程组x-1=3py^2，x^2＋x＋1=3z^2仅有正整数解（p，x，y，z）=（7，22，1，13）。     

关于Diophantine m-tuples的个数(英文)

对于一个由m个正整数构成的数组,若其中任意两个正整数的乘积都等于一个完全平方数减去1,则称这个数组为Diophantine m-tuples.最近,Dujella获得了关于Diophantine2-tuples及3-tuples个数的渐进公式.本文改进了的相关结果.     

Diophantine definitions for some polynomial rings

The author gives Diophantine definitions for polynomial rings of characteristic p > 0 over the polynomial rings with constants in finite fields, and for rational integers over polynomial rings of characteristic 0. The author also investigates Diophantine relationships between other pairs of polynomial rings of characteristic p > 0.