Systems of Diagonal Equations Over p-Adic Fields

Abstract

Let [ ] be a [pfr ]-adic field, and consider the system F = ( F 1 ,…, F R ) of diagonal equations [formula here] with coefficients in [ ]. It is an interesting problem in number theory to determine when such a system possesses a nontrivial [ ]-rational solution. In particular, we define Γ*( k , R , [ ]) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in [ ] has a nontrivial [ ]-rational solution provided only that N [ges ]Γ*( k , R , [ ]). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R , [ ]) = R + 1 for any field [ ]. We also define Γ*( k , R ) to be the smallest integer N such that Γ*( k , R , ℚ p ) [les ] N for all primes p .