Diophantine Analysis: From Structured to Random

该奖项关注的是整数和某些方程的真实的解的随机与确定性之间的紧张关系，以及整数的素因子数量的虚幻奇偶性。在特殊情况下，这些特性可应用于设计有效的电路和算法。在项目的实际应用方面，PI预计，如果构建物理量子计算机，将使用与算术群相关的丢番图方程的丰富解来设计最佳效率的通用量子门。理解整数代数方程的解是数论的中心主题。这个项目的重点是方程有许多这样的解决方案。唯一已知的通用方法产生的解决方案是哈迪-利特尔伍德“圆法”适用于当有许多变量和解决方案是非常丰富的。这个项目关注的是整数和实数上的这类方程的解的丰富性的研究，当有很少的变量，特别是在临界尺寸。在新的工具，以取得进展的动力学仿射多项式态射群，从超越和随机高斯场理论的辅助多项式的方法。在研究这些问题和相关问题时，一个主题是确定性与随机性，它允许在某些特殊情况下设计各种最佳效率的通用量子门。这个主题的另一个方向是对一个数的素因子个数的奇偶性的理论和计算研究。这个奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The award is concerned with the tension between the random versus deterministic properties of integer and real solutions to certain equations as well as for the illusive parity of the number of prime factors of an integer. These features in special cases have applications to the design of efficient circuits and algorithms. In terms of practical applications of the project, the PI expects that the design of optimally efficient universal quantum gates using the richness of solutions to Diophantine equations associated with arithmetic groups will be used if a physical quantum computer is built.Understanding solutions of algebraic equations in integers is a central theme in the theory of numbers. The focus of this project is on equations which have many such solutions. The only general method known to produce solutions is the Hardy-Littlewood "circle method" which applies when there are many variables and the solutions are very abundant. This project is concerned with the study of the richness of solutions to such equations over the integers and the reals, when there are few variables and especially in the critical dimension. Among the new tools to be employed to make progress are, the dynamics affine polynomial morphism groups, the method of auxiliary polynomials from transcendence and the theory of random Gaussian Fields. A theme that is captured in the study of these and related problems, is determinism versus randomness and it allows one in certain special cases to design various optimally efficient universal quantum gates. Another direction to be pursued with this theme is the theoretical and computational study of the parity in the number of prime factors of a number.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.