The Decoupling Approach for Parsell-Vinogradov Systems

The Decoupling Approach for Parsell-Vinogradov Systems

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Ruixiang Zhang, IAS
Fine Hall 314

Parsell-Vinogradov (P-V) systems are "high dimensional" generalizations of Vinogradov systems. For any P-V system, there is an easy lower bound for the number of integer solutions inside a box. It is conjectured that this lower bound is also an upper bound with at most an N^{\varepsilon}-loss. Two "parallel" theories, namely efficient congruencing and decoupling, have been developed to attack counting questions for such systems. As a fruitful result, Wooley and Bourgain-Demeter-Guth proved this conjecture in the Vinogradov case. After the background I'll present the decoupling approach in this talk, highlighting more basic decoupling theory for the paraboloid and the moment curve (i.e. Vinogradov case). Then I'll explain some new phenomena in general P-V systems. We will mention why there are some very complicated new combinatorial difficulties and how we are able to address them. This leads to a proof of the conjecture in dimension 2. We also hope that we have already dealt with the main difficulty in higher dimensional cases. Joint work in progress with Shaoming Guo.