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A ‘Waring's problem’ for homogeneous forms

Published online by Cambridge University Press:  24 October 2008

W. J. Ellison
W. J. Ellison
Affiliation:
St Catharine's College, Cambridge
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Extract

Professor H. Davenport has raised the following question. Let f(X) be a homogeneous form in n variables X, of degree k, with coefficients in a field K. Is it possible to write f(X) in the form

where the Li(X) are linear forms over K and N depends only on K, n, and k? In particular, what can be said when K = R, the reals; K = Q, the rationals; K = C, the complex numbers ?

We shall show that when k is odd such a representation is always possible and we shall obtain an explicit bound for N. However, if k is even, no such result is possible when K = R or Q, even when we impose the obvious necessary condition that f(X) be positive semi-definite. In fact we shall exhibit a construction for obtaining forms f(X) which are positive definite but cannot be expressed in the form

r even, where φi(X) is a homogeneous form of degree s and sr = k. We shall determine precisely which definite forms can be expressed in the required form. Some natural generalizations of the problem are then considered.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 65 , Issue 3 , May 1969 , pp. 663 - 672
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

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