A formula for the Euler characteristic of a real algebraic manifold
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.
scientific article; zbMATH DE number 724663
scientific article