Small Zeros of Quadratic Forms Over Number Fields

Abstract

Let F be a nontrivial quadratic form in N variables with coefficients in a number field k and let A be a K x N matrix over k.We show that if the simultaneous equations F(-x.) = 0 and Ax = 0 hold on a subspace X of dimension L and L is maximal, then such a subspace X can be found with the height of X relatively small.In particular, the height of X can be explicitly bounded by an expression depending on the height of F and the height of A. We use methods from geometry of numbers over adele spaces and local to global techniques which generalize recent work of H. P. Schlickewei.Recently H. P. Schlickewei [8] has extended Cassels' result for k = Q in a different direction.Suppose that L > 1 is the largest integer such that the quadratic form F vanishes on some L dimensional rational subspace of QN.Then Schlickewei has shown that there exist L linearly independent vectors cx, c2,..., cl in QN such that F vanishes identically on the subspace spanned by {fi, c2,..., Cl}, andIn particular, there exists a vector cx ^ 0 in QN with F(cx) = 0 and (i.4) h(cx) /2L.