Real plane algebraic curves with asymptotically maximal number of even ovals

Abstract

It is known for a long time that a nonsingular real algebraic curve of degree 2k in the projective plane cannot have more than 7/2*k^2-9/4*k+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 tends to 7/4$ as k tends to infinity, where p is the number of even ovals of the curves. We also show that the same kind of result is valid dealing with odd ovals.