Abstract

Motivated by Lagrange's theorem, which states that every natural number can be expressed as the sum of at most four squares, in 1770, Waring posed a question if there is a positive integer s associated with each natural number k, such that every natural number can be expressed as the sum of at most s natural numbers raised to the power k. Then, in 1900, Hilbert inquired about the solvability of a system of Diophantine equations of Waring type, now known as the Hilbert-Kamke system. A multi-dimensional version of these problems naturally arises when solving the same system in the polynomial rings over integers. One approach to these questions involves the Hardy-Littlewood circle method. To ensure that the major arc from the circle method provides the main contribution, it is necessary to establish an absolute positive lower bound of the singular series, which is a product of the densities of local solutions. This presentation will focus on the lower bound problem for the multi-dimensional Hilbert-Kamke system by examining its local solvability. We will specifically provide explicit bounds on the solvability over p-adic fields. It is a joint work with Yu-Ru Liu, Xiaomei Zhao, and Xingchi Ruan.