Diophantine Analysis: From Structured to Random

该奖项涉及某些方程的整数解和实数解的随机性与确定性之间的紧张关系，以及整数的素因数数目的虚幻奇偶性。在特殊情况下，这些特征可应用于高效电路和算法的设计。在该项目的实际应用方面，PI预计，如果建立物理量子计算机，将使用与算术群相关联的丢番图方程的丰富解来设计效率最高的通用量子门。理解整数代数方程的解是数论中的一个中心主题。这个项目的重点是有许多这样的解的方程。唯一已知的产生解的一般方法是Hardy-Littlewood“圆法”，该方法适用于变量较多且解非常丰富的情况。本课题主要研究在变量很少的情况下，特别是在临界维上，这类方程在整数和实数上的解的丰富性。在取得进展的新工具中，有动力学仿射多项式态射群、超越辅助多项式方法和随机高斯场理论。在这些和相关问题的研究中捕捉到的一个主题是决定论与随机性，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。这个主题要追求的另一个方向是对一个数字的质数的数量的平等性进行理论和计算研究。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.