The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves

Details The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves

2003-01-22

arxiv.org/abs/math/0301245v1

Mathematics

Algebraic Geometry

13 pages, 3 figures

Scientific paper

The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on the number of empty ovals. To evaluate its rate we show its equivalence with the rate of growth of the number of trees half of whose vertices are leaves and evaluate the latter rate.

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