Asymptotic behaviour of Betti numbers of real algebraic surfaces

Abstract

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 $\frac{13}{18} \leq l\_1 \leq \frac{5}{6}$.