Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-27T20:43:33.411Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.