A formula for the Euler characteristic of a real algebraic manifold
type: Item
instance of: scholarly article
title: A formula for the Euler characteristic of a real algebraic manifold
zbMATH Open document ID: 0824.14046
DOI: 10.1007/BF02568203
author: Zbigniew Szafraniec
published in: Manuscripta Mathematica
publication date: 1995-11-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156027
review text: Let \(F = (F_ 1, \dots, F_ k) : \mathbb{R}^ n \to \mathbb{R}^ k\) be a real polynomial mapping and \(W_ \mathbb{R} = \{x \in \mathbb{R}^ n | F(x) = 0\}\), \(M = {\partial (F_ 1, \dots, F_ k) \over \partial (x_ 1, \dots, x_ k)}\). Define \(w = x^ 2_ 1 + \cdots + x^ 2_ n\), \(I \subset \mathbb{R} [x_ 1, \dots,x_ n]\) the ideal generated by \(F_ 1, \dots, F_ k\) and all the \((k + 1) \times (k + 1)\)-minors \({\partial (w, F_ 1, \dots, F_ k) \over \partial (x_{i_ 1}, \dots, x_{i_{k + 1}})}\). Consider the algebra \(A_ \mathbb{R} = \mathbb{R} [x_ 1, \dots, x_ n]/I\). We assume that \(\dim_ \mathbb{R} A_ \mathbb{R} < \infty\). Two bilinear symmetric forms \(\Phi\) and \(\Phi_ M : A_ \mathbb{R} \times A_ \mathbb{R} \to \mathbb{R}\) are defined in terms of \(F_ 1, \dots, F_ k\) such that if \(\Phi_ M\) is nondegenerate then \(W_ \mathbb{R}\) is an \((n - k)\)- dimensional manifold and \(\chi (W_ R) = ( - 1)^ k\cdot \text{sign}(\Phi)\) if \(n-k\) is odd, \(\chi (W_ R)=\text{sign}(\Phi)\) if \(n - k\) is even. Since \(w\) may not have nondegenerate critical points on \(W_ \mathbb{R}\), a function \(\widetilde \Omega : W_ \mathbb{R} \to \mathbb{R}\) with only nondegenerate critical points which uniformly approximates \(w\) on \(W_ \mathbb{R}\) in \(C^ 2\)-topology is used with standard Morse theory. Then a formula of D. Eisenbud and H. Levine about 0-dimensional complete intersection algebras over \(\mathbb{R}\) is used to prove the result.
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14F45
zbMATH DE Number: 724663
zbMATH Keywords: real algebraic sets
zbMATH Keywords: Euler-Poincaré characteristic
zbMATH Keywords: real polynomial mapping
reviewed by: Rajendra Gurjar
MaRDI profile type: Publication
cites work: Singularities of differentiable maps, Volume 2. Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous and revised by the authors and James Montaldi
cites work: Q4714016
cites work: Euler Characteristics of Real Varieties
cites work: On topological invariance of the Milnor number mod 2
cites work: An algebraic formula for the degree of a \(C^\infty\) map germ. Sur une inegalite a la Minkowski pour les multiplicites
cites work: Q3714169
cites work: Q4115815
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Q4075559
cites work: On the Euler characteristic of analytic and algebraic sets
cites work: On the Euler characteristic of complex algebraic varieties
cites work: On the Euler characteristic mod 2 of real projective varieties
cites work: The Euler characteristic of algebraic complete intersections.
cites work: Q4014100
cites work: Topological invariants of weighted homogeneous polynomials
cites work: Topological invariance of the Milnor number mod 2
OpenAlex ID: W2052585175
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description: scientific article; zbMATH DE number 724663