Sommes de puissances dans les corps finis

Yvette AMICE et Bruno KAHN
  • Année : 1992
  • Tome : 209
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
  • Class. Math. : 11T06
  • Pages : 115-136
  • DOI : 10.24033/ast.158

The higher levels of a field $F$, studied among others by Parnami, Agrawal, Rajwade and Revoy, are defined like the ordinary level, as the smallest number of $n$-th power summands necessary to represent $-1$, where $n$ is a power of 2. When $F$ is a finite field $\mathbb {F}_q$ ($q$ odd), the increasing sequence of higher levels stabilises from $h(q)$ on, where $h(q)$ is the dyadic valuation of $q-1$ ; its supremum is denoted here by $s(q)$. It appears that $s(q) = 2$ unless $q = p$ or $p^3$, with $p =\mathrm {char} F$, and $s(p^3) = 2$ or 3. If $q = p$, the Weil (or Jacobi) sums estimates imply that $s(p) = 2$ as soon as $p > 2^{4h(p)}$. However, computations performed up to $10^9$ hint that this bound is much too big in practice, and at least for small values of $h(p)$ ($h(p)\leq 7$), one has $s(p) = 2$ as soon as $p > 2^{2.72h(p)}$. Similarly, computations find no prime $p$ such that $s(p^3) = 3$ up to $p = 101 711 873$. We believe that these experimental results exemplify general phenomena, but so far have no theoretical explanation for them.

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