Sharp bounds for small moments of multidimensional Weyl sums
The study of Diophantine equations (i.e., integer solutions of polynomial equations and systems of equations) has a long history going back thousands of years. One of the most powerful techniques for analyzing Diophantine equations is a collection of ideas that expresses solutions in terms of integrals of functions called Weyl sums; one component of this method is a mean value for Weyl sums, known as Vinogradov’s Mean Value.
Recent collaborative work in this area has resulted in new methods for analyzing the integer solutions of the type of equations known as Vinogradov’s system. The results led to sharp estimates for the number of solutions of the system for a wide range of s (the number of variables in the system) and settled a longstanding conjecture on the subject, which had been open since the 1930s. Subsequent work proved sharp bounds and the main conjecture for such systems when s is large.
During his Center appointment Professor Ford will contribute in this area through his continuing collaboration with professors Scott Parsell (West Chester University) and Sean Prendiville and Trevor Wooley (University of Bristol, U.K.). Their project will focus on establishing the main conjecture for such systems when s is small. At the heart of their method will be the efficient congruencing innovation developed by Professor Wooley, its adaptation to the realm of small s in Vinogradov’s Mean Value, and its extension to the multidimensional setting. One of their first tasks is to create a new “p-adic” theory where equations are replaced by congruences.
Their research results are expected to have application to the theory of more general types of equations, to the distribution of prime numbers, and to other questions in number theory.