Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Quadratic forms between spheres and the non-existence of sums of squares formulae

Paul Y. H. Yiua1

a1 Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Y4, Canada

Hurwitz [6] posed in 1898 the problem of determining, for given integers r and s, the least integer n, denoted by r s, for which there exists an [r, s, n] formula, namely a sums of squares formula of the type

S0305004100066226_eqnU1

where S0305004100066226_inline1 are bilinear forms with real coefficients in S0305004100066226_inline2 and S0305004100066226_inline3. Such an [r, s, n] formula is equivalent to a normed bilinear map S0305004100066226_inline4 satisfying S0305004100066226_inline5. We shall, therefore, speak of sums of squares formulae and normed bilinear maps interchangeably.

(Received January 09 1986)