Search

Let F be a finite field of characteristic distinct from 2, f and g quadratic forms over F , dim ⁡ f = dim ⁡ g = n . A particular case of Chevalley's theorem claims that if n ≥ 5 , then f and g have a common zero. We give an algorithm, which establishes whether f and g have a common zero in the case n ≤ 4 . The most interesting case is n = 4 . In particular, we show that if n = 4 and det ⁡ ( f + t g ) is a squarefree polynomial of degree different from 2, then f and g have a common zero. We investigate the orbits of pairs of 4-dimensional forms ( f , g ) under the action of the group GL 4 ( F ) , provided f and g do not have a common zero. In particular, it turns out that for any polynomial p ( t ) of degree at most 4 up to the above action there exist at most two pairs ( f , g ) such that det ⁡ ( f + t g ) = p ( t ) , and the forms f , g do not have a common zero. The proofs heavily use Brumer's theorem and the Hasse–Minkowski theorem. [ABSTRACT FROM AUTHOR]

Abstract: Let F be a field of characteristic not 2, , . Let further G be the subgroup of generated by . For any put . Consider the complex where are all the quadratic extensions of F, containing in M, , . The homology group of this complex at -th term from the left is denoted by . Independently of charF we give examples of these complexes with nontrivial and prescribed and . Also similar examples concerning the group are constructed. [Copyright &y& Elsevier]