On partitions of finite vector spaces of low dimension over GF(2)

AbstractLet Vn(q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of Vn(q) is a partition of Vn(q) if every nonzero vector in Vn(q) is contained in exactly one subspace of P. If there exists a partition of Vn(q) containing ai subspaces of dimension ni for 1≤i≤k, then (ak,ak−1,…,a1) must satisfy the Diophantine equation ∑i=1kai(qni−1)=qn−1. In general, however, not every solution of this Diophantine equation corresponds to a partition of Vn(q). In this article, we determine all solutions of the Diophantine equation for which there is a corresponding partition of Vn(2) for n≤7 and provide a construction of each of the partitions that exist

Applications of patching to quadratic forms and central simple algebras

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.Comment: 48 pages; connectivity now required in the definition of rational group; beginning of Section 4 reorganized; other minor change