On a Reciprocal Tarry-Escott Problem, the Distribution of Roots of Polynomials Modulo a Composite, and Sieve Methods

DMS-0301282 Poonen，Bjorn M.(Croot，Ernest S.)摘要标题：关于倒数Tarry-Escott问题，模为复合的多项式的根的分布，和筛子方法提出者的研究项目有三个目标：他希望完成他关于Prouhet-Tarry-Escott问题的结果和方法的工作，他将在一篇或多篇论文中撰写；他计划继续并在一篇论文中写出他与一名本科生在密码学方面的工作；他希望继续开发一种新的筛子方法，用于确定薄整数集的素数。作者在Prouhet-Tarry-Escott问题上所做的工作给出了一种构造多变量丢番图方程组的解的新方法，并且该方法可能导致该领域中更多尚未解决的问题的解。密码学的工作集中在证明某些用于攻击公钥密码系统的算法是不容易改进的(例如科珀史密斯方法)。最后，提出者计划继续开发一种筛法来计算给定整数集中的素数的数目，这允许人们使用关于该集合的额外的分析信息(除了组合筛子使用的通常数据之外)，希望该方法将导致质数理论中的一个或另一个已知的、困难的、未解决的问题的解决。自古希腊时代以来，数学家们一直试图理解素数是如何间隔的；也就是说，当人们考虑越来越大的素数时，连续素数之间的距离是如何变化的？筛子方法是作为回答这类问题的理论工具而发展起来的；然而，有许多关于这种间隔的自然问题目前它们无法回答。提出者的研究目标之一是完成开发一种新的筛分方法，他希望用这种方法在这些悬而未决的问题上取得进展。作者在密码学方面的工作是基于本科生攻破RSA密码体制的研究，RSA密码体制是一种通过互联网发送安全数据的过程。作者(和学生)计划继续他在数论中的一个相关问题上的工作，解决这个问题将表明这种攻击方法不能有太大改进。最后，提出者计划继续他关于Prouhet-Tarry-Escott问题的工作。这个问题是数论中一个长期悬而未决的问题，它的提出者正在开发新的方法来解决它和其他类似的问题。

DMS-0301282Poonen, Bjorn M.(Croot, Ernest S.)AbstractTitle: On a Reciprocal Tarry-Escott Problem, the Distribution of Roots Of Polynomials Modulo a Composite, and Sieve MethodsThe Proposer's research project has three goals: He wishes to finish working on his results and methods on the Prouhet-Tarry-Escott problem, which he will write up in one or more papers;he plans to continue and to write up in a paper his work with an undergraduate on cryptology; and, he wishes to continue to develop a new sieve method for determining the number of primes in thin sets of integers. The proposer's work on the Prouhet-Tarry-Escott problem gives a new method for constructing solutions to certain systems of diophantine equations with many variables, and the method may lead to a solution of one of more unsolved problems in this area. The work on cryptology centers around showing that certain algorithms for attacking public-key cryptosystems, that find low-height roots of polynomials modulo an integer q (such as Coppersmith's method), cannot be easily improved.Finally, the proposer plans to continue developing a sieve methodfor counting the number of primes in a given set of integers, which allows one to use additional analytic information (besides the usual data used by thecombinatorial sieve) about the set, in the hopes that the method willlead to the solution of one or another known, difficult, unsolved problemsin prime number theory.Since the time of the ancient Greeks, mathematicians have been trying tounderstand how the prime numbers are spaced; that is, how does the distancebetween consecutive prime numbers vary as one considers larger and largerprimes? Sieve methods were developed as a theoretical tool for answeringthis type of question; however, there are many natural questions about such spacings that they currently cannot answer. One of the proposer'sresearch goals is to finish developing a new sieve method which he hopes to use to make progress on some of these unsolved problems. The proposer'swork on cryptology was motivated by research with an undergraduate ona certain method (Coppersmith's algorithm) for attacking the RSA cryptosystem, which is a procedure for sending secure data via the internet.The proposer (and student) plans to continue his workon a related problem in number theory, the solution of which would show thatthis method of attack cannot be much improved. Lastly, the proposer plansto continue his work on the Prouhet-Tarry-Escott problem. This problem is a longstanding unsolved question in number theory, and proposer isdeveloping new methods to make progress on it and other, similar problems.