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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 \(n\).
14F25 Classical real and complex (co)homology in algebraic geometry
14P10 Semialgebraic sets and related spaces
14J70 Hypersurfaces and algebraic geometry
Full Text: DOI
[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).
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