On the values of $\varGamma ^*(k,p)$ and $\varGamma ^*(k)$
Volume 191 / 2019
Acta Arithmetica 191 (2019), 67-80
MSC: Primary 11D72; Secondary 11D79, 11D88.
DOI: 10.4064/aa180613-4-1
Published online: 29 July 2019
Abstract
For $k\in\mathbb{N}$ and $p$ a prime number, define $\varGamma^*(k,p)$ to be the smallest $n \in \mathbb{N}$ such that every diagonal form $a_{1}x_{1}^k + \cdots + a_{s}x_{s}^{k}$ with integer coefficients has a nontrivial zero over $\mathbb{Q}_p$ whenever $s\geq n$. Define also $$\varGamma^{*}(k) = \displaystyle \max _{p \,{\rm prime}} \varGamma^*(k,p).$$ We prove an upper bound for $\varGamma^*(k,p)$ and show that it is equal to $\varGamma^*(k,p)$ whenever $p-1$ divides $k$. We also find the exact value of $\varGamma^*(54)$.
