Exact values of the function \(\varGamma ^{*}(k)\)
The function \(\Gamma^*(k)\) is the least integer \(s\) such that any diagonal form of degree \(k\) over \(\mathbb{Q}\) in \(s\) variables has a non-trivial \(p\)-adic zero, for every prime \(p\). \textit{H. Davenport} and \textit{D. J. Lewis} [Proc. R. Soc. Lond., Ser, A 274, 443--460 (1963; Zbl 0118.28002)] showed that \(\Gamma^*(k)\leq k^2+1\) for all \(k\geq 1\), with equality when \(k+1\) is prime. In principle there is a standard finite computation to calculate \(\Gamma^*(k)\) for any individual \(k\), but until recently only the values for \(k\leq 11\), together with those for which \(k+1\) is prime, had been handled. The author, in work to appear, has improved the situation by treating odd \(k\leq 25\). The present paper evaluates \(\Gamma^*(k)\) for all \(k\leq 31\), and deals with the degree 32 case for all \(p\)-adic fields with \(p\not=2\). The proofs handle the various values of \(k\) separately, using a variety of techniques to reduce as much as possible the amount of computational testing required.
scientific article; zbMATH DE number 5931163