Real plane algebraic curves with asymptotically maximal number of even ovals

Abstract

It has been known for a long time that a nonsingular real algebraic curve of degree $2k$ in the projective plane cannot have more than $7k^2/4-9k/4+3/2$ even ovals. We show here that this upper bound is asymptotically sharp; that is to say, we construct a family of curves of degree $2k$ such that $p/k^2\to {}_{k\to \infty }7/4$, where $p$ is the number of even ovals of the curves. We also show that the same kind of result is valid when dealing with odd ovals