By Beniamino Segre
ISBN-10: 3642527647
ISBN-13: 9783642527647
ISBN-10: 3642527663
ISBN-13: 9783642527661
Read Online or Download Some Properties of Differentiable Varieties and Transformations: With Special Reference to the Analytic and Algebraic Cases PDF
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Extra info for Some Properties of Differentiable Varieties and Transformations: With Special Reference to the Analytic and Algebraic Cases
Example text
17. Projective invariants of two curves having the same osculating spaces at a point. - Let us consider, in the plane, two branches C, C' of analytic curves having the same centre 0 and the same tangent t. On the hypothesis that the two branches can be represented simultaneously in the form y=axl'+···, y=a'x"+'" (aa' =l= 0, and {l an integer or fraction> 1) (for example, if {l is an integer, this condition is equivalent to saying that the two branches have at 0 the same order of contact, {l, with t; and it is not difficult to specify the geometrical equivalent of this algebraic restriction in general), then the number a/a' is independent of the coordinate system and so constitutes a projective invariant of C, C' at 0, called a MEHMKE-SEGRE invariant.
Z 1 (5') (analogous to (5)), where the coefficients aii . l' bij ... /' ... , cii . 1 are positive real constants, for which it suffices to observe that ai j ••• 1 = bij ... 1 = ... = cii ... 1 ~ e()k-I In. Taking the equations of T in the form (5), we see from the convergence of these series that there exist positive real constants cp, u such that, for any choice of the indices i, j, .. ,1 satisfying (15), we have laii ... II ~ u cp-k, Ib ii ... 11 ~ u cp-k, ... , ICii ... 11 ~ u cp-k . L) = Ie min (' 1, nu (Ie> 0) , we have, for any integer k ;;;; 2, lek-I-S:; - eIP • nu It follows that laij ...
And the same restriction is found in the researches of FATou [36J, concerning the functional equations (generalizing those of SCHRODER) which appear in such investigations. The case considered in the final proposition of § 16 thus goes well beyond the work of these authors; and it shows what serious difficulties would be presented, should we wish to complete the classification of LATTES [54, 55J for the reduced forms of analytic transformations in two variables in the neighbourhood of a united point.
Some Properties of Differentiable Varieties and Transformations: With Special Reference to the Analytic and Algebraic Cases by Beniamino Segre
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