Solvability of systems of diagonal equations over p‐adic local fields.

We prove that a system of R diagonal equations of degree d over a finite extension K of Qp has a non‐trivial solution in K if the number of variables exceeds 3R2d2 (if p>2) or 8R2d2 (if p=2). As a consequence, a system of R homogeneous equations of degree d over K has a non‐trivial solution in K if the number of variables exceeds (8R2d2)2d−1. [ABSTRACT FROM AUTHOR]