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

  • Paul Y. H. Yiu (a1)

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

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

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[1]J. F. Adams . On the nonexistence of elements of Hopf invariant one. Ann. of Math. 72 (1960), 20104.

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[19]E. Stiefel . Üjer Richtungsfelder in den projektiven Raumen und einen Satz aus reellen Algebra. Comment. Math. Helv. 13 (1941), 201218.

[21]R. Wood . Polynomial maps from spheres to spheres. Invent. Math. 5 (1968), 163168.

[24]S. Yuzvinsky . A series of monomial pairings. Linear and Multilinear Algebra 15 (1984), 109119.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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