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Homogeneous forms of odd degree in a large number of variables

Published online by Cambridge University Press:  26 February 2010

B. J. Birch
B. J. Birch
Affiliation:
Trinity College, Cambridge.
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Extract

It is well known that an indefinite quadratic form with integral coefficients in 5 or more variables always represents zero properly, and this has raised the problem of proving a similar result for forms of higher degree, namely that such a form, of degree r, represents zero properly if the number of variables exceeds some number depending only on r. For a form of odd degree, no condition corresponding to indefiniteness is needed, but for a form of even degree (4 or more) some even stronger condition must be required.

Type
Research Article
Information
Mathematika , Volume 4 , Issue 2 , December 1957 , pp. 102 - 105
Copyright
Copyright © University College London 1957

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References

1.Brauer, R., “Systems of homogeneous algebraic equations”, Bull. American Math. Soc. (2), 51 (1945), 749–755.CrossRefGoogle Scholar
2.Peck, L. G., “Diophantine equations in algebraic number fields”, American J. of Math., 71 (1949), 387–402.CrossRefGoogle Scholar