turkmath.org

Consider a real algebraic curve $A$ in a real algebraic surface $X$ (typically, rational) and assume that the set $\mathbb{R}A$ of the real points of $A$ is finite. (Certainly, this implies that the class $[A]$ of $A$ in $H_2(X)$ is even and each point of $\mathbb{R} A$ is a singular point of $A$.) Recently, quite a few researchers showed considerable interest in the possible cardinality of the finite set $\mathbb{R}A$. We give a partial answer (upper and lower bounds) to this question in terms of either the class $[A]$ alone or the class $[A]$ and genus $g(A)$; in the latter case, our bounds are often sharp. In the simplest case where $A\subset\mathbb{P}^2$ is a plane curve of degree $2k$, we have $|\mathbb{R} A|\le k^2+g(A)+1$ (sharp if $g(A)$ is small compared to $k$) and $|\mathbb{R} C|\le \frac{3}{2}k(k-1) + 1$ (sharp for $1\le k\le4$ but, most likely, not sharp in general). I will discuss the proof of the upper bounds (essentially, Petrovsky's inequality) and a few simple constructions for the lower bounds.