We prove that the essential dimension of the spinor group Spin_n grows
exponentially with n; in particular, we give a precise formula for this
essential dimension when n is not divisible by 4. We use this result to show
that the number of 3-fold Pfister forms needed to represent the Witt class of a
general quadratic form of rank n with trivial discriminant and Hasse-Witt
invariant grows exponentially with n.
This paper overlaps with our earlier preprint arXiv:math/0701903 . That
preprint has splintered into several parts, which have since acquired a life of
their own. In particular, see "Essential dimension of moduli of curves and
other algebraic stacks", by the same authors, and "Some consequences of the
Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).Comment: 11 pages. Accepted for publication in Annals of Mathematic
