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$evenovals.Weshowherethatthisupperboundisasymptoticallysharp,thatistosayweconstructafamilyofcurvesofdegree2ksuchthatp/k^{2}tendsto7/4$ 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.

12 pages, 10 figures

References in corpus (1)

• Asymptotiques de nombres de Betti d'hypersurfaces projectives réelles