Bounds for homology groups of some manifolds (Q1335893)

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 \(n\).