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A general canonical expression

Published online by Cambridge University Press:  24 October 2008

J. Bronowski
J. Bronowski
Affiliation:
Jesus College
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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

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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