Asymptotic behaviour of Betti numbers of real algebraic surfaces
Frédéric Bihan · Commentarii Mathematici Helvetici · 2003
Let X_m be a nonsingular real algebraic surface of degree m in the complex projective space {\mathbb C}P^3 and {\mathbb R}X_m its real point set in {\mathbb R}P^3 . In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value \beta_{i,m} of the Betti number b_i({\mathbb R}X_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} .