Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-30T19:23:34.844Z Has data issue: false hasContentIssue false
  • English
  • Français

A general canonical expression

Published online by Cambridge University Press:  24 October 2008

J. Bronowski
J. Bronowski
Affiliation:
Jesus College
Get access Rights & Permissions [Opens in a new window]

Extract

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:

Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that

(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;

(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 4 , 30 October 1933 , pp. 465 - 469
Copyright
Copyright © Cambridge Philosophical Society 1933

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

* Bronowski, J., Proc. Camb. Phil. Soc. 29 (1933), 69.CrossRefGoogle Scholar

† See, for example, Wakeford, E. K., Proc. London Math. Soc. (2), 18 (1919), 403Google Scholar; or J. Bronowski, loc. cit. 75.

* This condition can also be stated in the form: any h tangent spaces of V lie together in a prime of N. In this form, this theorem (i) is a generalisation of the theorem of Terracini, A., Rend. di Palermo, 31 (1911), 392CrossRefGoogle Scholar. It may also be interestingly compared with a theorem of Segre, C., Atti Acc. Torino, 42 (1906–1907), 1047Google Scholar, which is in effect a discussion of the limiting case h = 1, when r = 2, k = 1.

* Bronowski, J., Journ. London Math. Soc., 8 (1933), 308–312.CrossRefGoogle Scholar