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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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##### References:
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