Karpushkin, V. N. Bounds for homology groups of some manifolds. (English. Russian original) Zbl 0812.14012 Funct. Anal. Appl. 27, No. 2, 146-147 (1993); translation from Funkts. Anal. Prilozh. 27, No. 2, 86-87 (1993). Given the hypersurface \(F=0\) for a degree \(n\) polynomial \(F \in \mathbb{C}^ k [x]\), the author considers the projective closure of \(\{\text{Re} F = 0\}\), named \(\Gamma\), and of \(\{\text{Re} F \geq 0\}\), named \(L\). He gives upper bounds for the \(\mathbb{Z}_ 2\)-dimension of \(H_ * (\Gamma, \mathbb{Z}_ 2)\) and for the Euler characteristic \(\chi (L)\) which are of order \(n^ k\) for odd \(n\) and of order \(n^{2k-1}\) for even \(n\). Reviewer: G.E.Winkler (Berlin) MSC: 14F25 Classical real and complex (co)homology in algebraic geometry 14P10 Semialgebraic sets and related spaces 14J70 Hypersurfaces and algebraic geometry Keywords:homology groups; hypersurface PDF BibTeX XML Cite \textit{V. N. Karpushkin}, Funct. Anal. Appl. 27, No. 2, 146--147 (1993; Zbl 0812.14012); translation from Funkts. Anal. Prilozh. 27, No. 2, 86--87 (1993) Full Text: DOI References: [1] I. G. Petrovskii and O. A. Oleinik, Izv. Akad. Nauk SSSR, Ser. Mat.,13, 389-402 (1949). [2] O. A. Oleinik, Mat. Sb.,28, 635-640 (1951). [3] D. A. Gudkov, Usp. Mat. Nauk,29, No. 4, 3-79 (1974). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.