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 $\frac{13}{36}\le l_0\le \frac{5}{12}$ and $\frac{13}{18}\le l_1\le \frac{5}{6}$.