Diophantine Analysis: From Structured to Random

Princeton University, Princeton NJ

Investigators

Abstract

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.