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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 $$n$$.
##### 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
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##### 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).
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