UC Davis

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov’s mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument.

Let be a natural number and . Define the exponential sum For , consider mean values of the exponential sums where we wrote . By making use of the aforementioned tools, we obtain the sharp upper bound for , for and . Furthermore, for , we obtain analogous results depending on a small cap decoupling inequality for the moment curves in .