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Sums of Squares Formulae With Integer Coefficients

Published online by Cambridge University Press:  20 November 2018

Paul Y. H. Yiu
Paul Y. H. Yiu*
Affiliation:
Department of Mathematics University of British Columbia Vancouver, B.C. Canada
*
Current address: Department of Mathematics Ohio State University Columbus, Ohio 43210, USA
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Abstract

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Hidden behind a sums of squares formula are other such formulae not obtainable by restriction. This drastically simplifies the combinatorics involved in the existence problem of sums of squares formulae, and leads to a proof that the product of two sums of 16 squares cannot be rewritten as a sum of 28 squares, if only integer coefficients are permitted. We also construct all [10, 10, 16] formulae.

Keywords

Sums of squares formulaeintercalate matricesstable homotopy classes of spheresnonsingular bilinear mapsHopf-Stiefel conditionPrimary 11E25Secondary 55Q4505B99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 30 , Issue 3 , 01 September 1987 , pp. 318 - 324
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Adem, J., Construction of some normed maps, Bol. Soc. Mat. Mexicana 20 (1975), pp. 59—75.Google Scholar
2. Cayley, A., On the theorem of the 2, 4, 8 and 16 squares. Quart. J. Math. 17 (1881), pp. 258–276; reprinted in Collected Papers 763.Google Scholar
3. Dickson, L.E., On quaternions and their generalization and the history of the eight square theorem, Annals of Math. 20 (1919), pp. 155–171.Google Scholar
4. Lam, K.Y., Sectioning vector bundles over real projective spaces, Quart. J. Math. Oxford (ser. 2) 23(1972), pp. 97–106.Google Scholar
5. Lam, K.Y., Some interesting examples of nonsingular bilinear maps, Topology 16 (1977), pp. 185–188.Google Scholar
6. Lam, K.Y., Topological methods for studying the composition of quadratic forms, Canadian Math. Soc. Conf. Proc. 4 (1984), pp. 173–192.Google Scholar
7. Lam, K.Y., Some new results in composition of quadratic forms, Invent. Math. 79 (1985), pp. 467–474.Google Scholar
8. Lam, K.Y. and Yiu, P., Retrieving hidden sums of squares formulae; preprint.Google Scholar
9. Milgram, R.J., Immersing projective spaces, Annals of Math. 85 (1967), pp. 473—482.Google Scholar
10. Milgram, R.J., Strutt, J. and Zvengrowski, P., Projective stable stems of spheres, Bol. Soc. Mat. Mexicana 22 (1977), pp. 48–57.Google Scholar
11. Paechter, G.F., The groups πr(vn.m), Quart. J. Math. Oxford 7 (1956), pp. 249–268.Google Scholar
12. Yiu, P., Thesis, University of British Columbia, 1985.Google Scholar
13. Yiu, P., Quadratic forms between spheres and the nonexistence of sums of squares formulae, Math. Proc. Cambridge Philos. Soc. 100 (1986), pp. 493–504 Google Scholar
14. Yuzvinsky, S., Orthogonal pairings of euclidean spaces, Michigan J. Math. 28 (1981), pp. 131 — 145.Google Scholar