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The addition of norm forms

Published online by Cambridge University Press:  26 February 2010

B. J. Birch ,
H. Davenport  and
D. J. Lewis
B. J. Birch
Affiliation:
Trinity College and Churchill College, Cambridge.
H. Davenport
Affiliation:
Trinity College and Churchill College, Cambridge.
D. J. Lewis
Affiliation:
Trinity College and Churchill College, Cambridge.
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Extract

Several recent papers † have been devoted to the problem of the solubility in integers of a homogeneous equation (or of simultaneous equations), or the representation of an integer by a form (homogeneous polynomial). Such problems can be regarded as extensions of Waring's Problem: that of representing an integer as x1k+…+x8k with positive integral x1 …, x8. The methods used are developments of the Hardy-Littlewood method, or if not they employ auxiliary results proved by that method. The number of variables needed to ensure success is usually very large when the degree k of the homogeneous form is large. In this note we draw attention to a somewhat special problem for which a comparatively small number of variables, namely 2k+1, suffices.

Type
Research Article
Information
Mathematika , Volume 9 , Issue 1 , June 1962 , pp. 75 - 82
Copyright
Copyright © University College London 1962

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References

† See Birch, B. J., “Forms in many variables”, Proc. Royal Soc. A, 265 (1962), 245–263.CrossRefGoogle Scholar

‡ By a theorem of Stouff (Bachmann, Quadratische Formen II, Kap. 11) any form of degree k in k variables with integral coefficients, which factorizes into real or complex linear forms but is irreducible over the rational field, is expressible as in (1) but with any linearly independent integers of the field in place of ω1 …, ωk, and with possibly a numerical factor. Our results would extend to such general norm forms.

page77 note3 † The symbol < indicates an inequality with an unspecified factor which is independent of P.

page78 note4 † See Vinogradov, I. M., Bull. Acad. Sci. U.B.S.S. (6), 21 (1927), 567–578Google Scholar, or Landau, E., Math. Zetischrift, 31 (1930), 319–338.CrossRefGoogle Scholar

page80 note5 † “Waring's problem in algebraic number fields”, Proc. Cambridge Phil. Soc., 57 (1961), 449–459.CrossRefGoogle Scholar

page81 note6 † Landau, , Vorleeungen über Zahlentheorie, I (Leipzig, 1927), Satz 315.Google Scholar

page82 note7 † Birch, , loc. cit., Lemma 7.1.Google Scholar