Speaker: Xiaochun Li
Title: Mean value inequalities related to Waring’s problem
Abstract: Waring’s problem is a long standing problem in analytic number theory, asking for the smallest value of $s$ such that every natural number is the sum of at most $s$ natural numbers raised to the power $k$. Such a problem can be reduced to a sharp $L^p$-estimate for certain mean value of the exponential sum. The mean values inequalities can be further deduced to a decoupling estimate in Fourier analysis.
A weak version of the decoupling estimates can be obtained by using the polynomial partitioning method.
Moreover, we notice that a sharp estimate of an incomplete exponential sum over finite fields can be used to make a sharp estimate at the critical $p$.
