Cardiff University

On vectors whose span contains a given linear subspace I. Aliev∗(Vienna), A. Schinzel (Warsaw) and W. M. Schmidt (Boulder, Co) Estimates are given for the product of the lengths of integer vectors spanning a given linear subspace. 2000 Mathematics Subject Classification: 11H06, 11H50 Key words: lattice, height of a subspace, successive minima The aim of this paper is to estimate for k> l> m> 0 c(k, l,m) = sup infH(S) l−k k−m l∏ i=1 |pi | , (1) where the supremum is taken over all subspaces S of Qk of dimension m and the infimum is taken over all sets of linearly independent vectors p1,...,pl in Zk, whose span contains S. Here H(S) is the determinant of the lattice S ∩ Zk and |p | is the Euclidean norm of p. Let γr,s be the generalized Hermite constant, as defined by Rankin [7], i. e. the least number such that every lattice Λ of rank r in Rr has a sublattice Γ of rank s and determinant det Γ ≤ γ1/2r,s (det Λ)s/r. γr,1 = γr,r−1 = γr is the ordinary Hermite constant. We shall prov