On common zeros of a pair of quadratic forms over a finite field

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 + tg) 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 + tg) = 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. (C) 2018 Elsevier Inc. All rights reserved.