Nesterenko's linear independence criterion for vectors

Abstract

In this paper we deduce a lower bound for the rank of a family of \(p\) vectors in \(\R^k\) (considered as a vector space over the rationals) from the existence of a sequence of linear forms on \(\R^p\), with integer coefficients, which are small at \(k\) points. This is a generalization to vectors of Nesterenko's linear independence criterion (which corresponds to \(k=1\)), used by Ball-Rivoal to prove that infinitely many values of Riemann zeta function at odd integers are irrational. The proof is based on geometry of numbers, namely Minkowski's theorem on convex bodies.