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Euler Characteristics of Real Varieties
Author(s) -
Bruce J. W.
Publication year - 1990
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/22.6.547
Subject(s) - mathematics , euler characteristic , degree (music) , singularity , algebraic variety , variety (cybernetics) , polynomial , algebraically closed field , projective variety , euler's formula , algebraic number , pure mathematics , isolated singularity , algebraic geometry , mathematical analysis , statistics , physics , acoustics
In this paper we show how to associate to any real projective algebraic variety Z ⊂ R P n −1 a real polynomial F 1 : R n ,0 → R , 0 with an algebraically isolated singularity, having the property that χ( Z ) = ½(1 − deg (grad F 1 ), where deg (grad F 1 is the local real degree of the gradient grad F 1 : R n , 0 → R n , 0. This degree can be computed algebraically by the method of Eisenbud and Levine, and Khimshiashvili [5]. The variety Z need not be smooth. This leads to an expression for the Euler characteristic of any compact algebraic subset of R n , and the link of a quasihomogeneous mapping f : R n , 0 → R n , 0 again in terms of the local degree of a gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitrary algebraic subset of R n and the link of any polynomial map are given in terms of the degrees of algebraically finite gradient maps. These maps do involve ‘sufficiently small’ constants, but the degrees involved ar (theoretically, at least) algebraically computable.