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Computation of the Euler characteristic of intersections of real quadrics. (English. Russian original) Zbl 0721.55001

Sov. Math., Dokl. 37, No. 2, 297-300 (1988); translation from Dokl. Akad. Nauk SSSR 299, No. 1, 11-14 (1988).
The authors consider not only quadratic equations but also quadratic inequalities. Namely, let a homogeneous quadratic map p: \({\mathbb{R}}^ N\to {\mathbb{R}}^ k\) together with a convex closed cone K in \({\mathbb{R}}^ k\) be given. It is assumed that p is regular with respect to K, i.e. there is no pair \(x\in {\mathbb{R}}^ N\setminus 0\), \(\omega \in K^*\setminus 0\), where \(K^*=\{\omega \in {\mathbb{R}}^{k*}:\omega\) \(y\leq 0\) for all \(y\in K\}\) is the cone dual to K, such that both conditions \(\omega p(x)=0\), p(x)\(\in K\) are satisfied. The authors’ Theorem 1 gives the formula \(\chi (\pi^{-1}K)=(1+(-1)^{N-1})/2-\sum^{N-1}_{n=0}(-1)^ n\chi (\Omega_ n)\) where \(\pi\) : \({\mathbb{R}}P^{N-1}\to {\mathbb{R}}^ k\) is determined by restricting p to the unit sphere in \({\mathbb{R}}^ N\), \(\chi\) is the Euler characteristic and \(\Omega_ n\) is the subset of \(K^*\) formed by elements \(\omega\) of unit length such that the diagonal representation of \(\omega\) p has at least N-n positive squares. The paper contains some related formulas and some applications to determination of boundary points for images of nonlinear maps.

MSC:

55M99 Classical topics in algebraic topology
14F45 Topological properties in algebraic geometry
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