Asymptotic behaviour of Betti numbers of real algebraic surfaces (Q1405746)

Summary: Let \(X_m\) be a non-singular real algebraic surface of degree \(m\) in the complex projective space \(\mathbb C\mathbb P^3\) and \(\mathbb RX_m\) its real point set in \(\mathbb R\mathbb P^3\). In the spirit of the sixteenth Hilbert's problem, one can ask, for each degree \(m\), for the maximal possible value \(\beta_{i,m}\) of the Betti number \(b_i(\mathbb RX_m)\) (\(i=0\) or 1). We show that \(\beta_{i,m}\) is asymptotically equivalent to \(l_i\cdot m^3\) for some real number \(l_i\) and prove inequalities \(\frac{13}{36} \leq l_0 \leq \frac{5}{12}\) and \(\frac{13}{18} \leq l_1 \leq \frac{5}{6}\).