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On the values of $\varGamma ^*(k,p)$ and $\varGamma ^*(k)$

Volume 191 / 2019

Hemar Godinho, Michael P. Knapp, Paulo H. A. Rodrigues, Daiane Veras 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 $\varGamma^*(54)$.

Authors

  • Hemar GodinhoDepartamento de Matemática
    Universidade de Brasília
    Brasília, DF 70910-900, Brazil
    e-mail
  • Michael P. KnappDepartment of Mathematics and Statistics
    Loyola University Maryland
    Baltimore, MD 21210-2699, U.S.A.
    e-mail
  • Paulo H. A. RodriguesInstituto de Matemática e Estatística
    Universidade Federal de Goiás
    Goiânia, GO 74690-900, Brazil
    e-mail
  • Daiane VerasInstituto federal de Goiás
    Avenida Saia Velha, Km 6, BR-040, s/n
    Parque Esplanada V
    Valparaíso de Goiás, GO 72876-601, Brazil
    e-mail

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