Real plane algebraic curves with asymptotically maximal number of even ovals

Details Real plane algebraic curves with asymptotically maximal number of even ovals Real plane algebraic curves with asymptotically maximal number of even ovals

2004-11-04

arxiv.org/abs/math/0411097v1

Mathematics

Algebraic Geometry

12 pages, 10 figures

Scientific paper

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.

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