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Hypersurfaces Fixed by Equiaffinities

Published online by Cambridge University Press:  20 November 2018

J. C. Fisher
J. C. Fisher*
Affiliation:
University of Saskatchewan, Regina, Saskatchewan
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In this note we state and prove the following

Any equiaffinity acting on the points of an n-dimensional vector space (n ≥2) leaves invariant the members of a one parameter family of hypersurfaces defined by polynomials p(xl…,xn)=c of degree m ≤n.

The theorem, restricted to the real plane, appears to have been discovered almost simultaneously by Coxeter [4] and Komissaruk [5]. The former paper presents an elegant geometric argument, showing that the result follows from the converse of Pascal's theorem. The present approach is more closely related to that of [5], in which the transformations are reduced to a canonical form.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 1 , March 1973 , pp. 129 - 131
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Artzy, R., Linear geometry, Addison-Wesley, Reading, Mass., 1965.Google Scholar
2. Birkhoff, G., andMacLane, S., A survey of modern algebra, Macmillan, New York, 1965.Google Scholar
3. Coxeter, H.S.M., Introduction to geometry, 2nd ed., Wiley, New York, 1969.Google Scholar
4. Coxeter, H.S.M., Affinely regular polygons, Abh. Math. Sem. Univ. Hamburg, 34 (1969), 38–58.Google Scholar
5. Komissaruk, A.M., The foundations of affine geometry in the plane [Osnovi Affinoĭ Geometrĭ na Ploskosti ]. Izdat. Vysš Skola, Minsk, 1967.Google Scholar