Bounds for the representation of quadratic forms

Abstract

We prove first that, for fixed integers n, m⩾1, there is a uniform bound on the number of Pfister forms of degree n over any Pythagorean field F necessary to represent (in the Witt ring of F) any form of dimension m as a linear combination of such forms with non-zero coefficients in F. “Uniform” means that the bound does not depend either on the form or on the field F; it is given by a recursive function f of n and m. Similar results hold for the reduced special groups arising from preordered fields and from fields whose Pythagoras number is bounded by a fixed integer. We single out a large class of Pythagorean fields and, more generally, of reduced special groups (cf. [4]) for which f has a simply exponential bound of the form cm n−1 ( c a constant). Such a class is closed under certain—possibly infinitary—operations which preserve Marshall's signature conjecture. In the case of groups of finite stability index s, we obtain an upper bound for f which is quadratic on [ m/2 n ], where c depends on s.