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On a method of Mordell in the geometry of numbers

Published online by Cambridge University Press:  26 February 2010

J. V. Armitage
J. V. Armitage
Affiliation:
University College, London.
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Extract

In several of his papers, Mordell has developed a method which, in certain cases, leads to an inequality connecting the critical determinant of an n dimensional star body with that of a related n—1 dimensional star body. The purpose of this paper is to exhibit the underlying principle in a general form and to show that the same principle can sometimes be carried further.

Type
Research Article
Information
Mathematika , Volume 2 , Issue 2 , December 1955 , pp. 132 - 140
Copyright
Copyright © University College London 1955

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References

page 132 note † The principal references are: J. London Math. Soc., 17 (1942), 107–15Google Scholar; Recueil Mathématique (Moscow), 12 (54) (1943), 273–6Google Scholar; J. London Math. Soc., 19 (1944), 3–6CrossRefGoogle Scholar; J. London Math. Soc., 19 (1944), 6–12CrossRefGoogle Scholar.

page 132 note ‡ We specify the first r coordinates here merely for convenience of formulation.

page 132 note § The adjoint of a real linear transformation from x1, …, xn to x1′, …, xn′ is the transformation from y1, …, yn to y1′, …, yn′ for which

identically.

page 132 note ¶ If r = 1, the left-hand side of (2) is to be read as F(0, x2, …, xn).

page 133 note † In describing particular star bodies it is convenient not to insist on the function on the left being homogeneous of degree 1.

page 134 note † In view of a theorem of Mahler we could actually take λ to satisfy (11) with equality but we do not need this fact.

page 136 note † Mordell's result actually included the case when r of the forms are real and there are s pairs of conjugate complex forms. This case is covered by the result (18) of (iii), applied to the region {x 1 … x r(y 12+z 12) … (y s2+z 12)} < 1. The remainder of the argument is then as above.

page 139 note † Journal London Math. Soc., 30 (1955), 186–195Google Scholar. For the present purpose we really only need that C3 <(Δ(K))-1, the proof of which is quite simple.

page 139 note ‡ London Ph.D. dissertation, 1956. The method used is similar to that developed by Mordell in Trans. American Math. Soc., 59 (1946), 189–215CrossRefGoogle Scholar for a class of plane regions, but complications arise because one of the essential conditions imposed by Mordell is not satisfied here.