Welcome to Pure Mathematics

The Hilbert-Kamke problem is a generalization of Waring’s problem to a system of equations, in which the sums of the first n powers of the same variables are given. By the circle method, the existence of solutions depends on a positive lower bound for the singular series. In this talk, we study this problem over a number field. We set up the circle method, following Siegel and Wang, and write the singular series as a product of local densities. Under a necessary arithmetic condition, and with enough variables, we prove that the singular series is bounded below by a positive constant depending only on the number of equations and the degree of the field. The proof is given first for quadratic fields and then for every number field.

MC 5501