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 7k2/4-9k/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/k2→k→∞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