Bounds for the Betti numbers of a level surface of a harmonic polynomial (Q1320686)

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 \(\text{rank} H_ 0 (V) \leq 2g + 2\).