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Bounds for the Betti numbers of a level surface of a harmonic polynomial. (English. Russian original) Zbl 0799.14032
Funct. Anal. Appl. 26, No. 4, 300-302 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 86-88 (1992).
Let \(\Pi_ 4(n)\) be the Petrovskii number in \(\mathbb{R} \mathbb{P}^ 3\): \(\Pi_ 4 (n) = (n -1)^ 3 - n(n - 1) (n-2)/3\). Denote by \(g = c^ 2_{n - 1}\) the genus of a nonsingular curve of degree \(n\) in \(\mathbb{C} \mathbb{P}^ 2\).
Theorem 1. Let \(W\) be the closure in \(\mathbb{R} \mathbb{P}^ 3\) of a level surface of a harmonic polynomial of degree \(n\) in \(\mathbb{R}^ 3\); then \(\text{rank} H_ 0 (W) \leq g + 1\). – If we assume that \(W\) is nonsingular, then \(\text{rank} H_ 1(W) \leq \Pi_ 4 (n) + 3 + 2g\). If in addition \(n \geq 13\), then \(W\) is not an \(M\)-manifold.
Theorem 2. Let \(V\) be a level surface of a harmonic polynomial of degree \(n\) in \(\mathbb{R}^ 3\), then \(\text{rank} H_ 0 (V) \leq 2g + 2\).

MSC:
14N05 Projective techniques in algebraic geometry
14J25 Special surfaces
14F45 Topological properties in algebraic geometry
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