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Real algebraic curves, the moment map and amoebas. (English) Zbl 1073.14555
From the text: A real algebraic projective curve $$C\subset \mathbb{R}\mathbb{P}^2$$ is the zero set of a polynomial $$p$$ of degree $$d$$. If it is nonsingular, it is homeomorphic to a disjoint union of circles. By Harnack’s inequality, $$C$$ has at most $$((d-1)(d-2)/2)+1$$ connected components; in the case of equality, $$C$$ is called an $$M$$-curve. We say that $$C$$ is in maximal position, of class $$n$$, with respect to lines $$l_1,\cdots,l_n$$ in $$\mathbb{R}\mathbb{P}^2$$, if (1) $$C$$ is an $$M$$-curve; (2) there exist arcs $$a_1,\cdots,a_n$$ in $$C$$ such that $$a_j$$ intersect $$l_j$$ in $$d$$ points and $$\bigcup_ia_i$$ is contained in the same component of $$C$$. The main results are the following.
Theorem 1. The maximal topological type is unique for $$n=3$$ and any $$d=\deg p$$.
Theorem 2. There are no maximal topological types for $$n>3$$ and $$d>3$$.

MSC:
 14P05 Real algebraic sets 14P25 Topology of real algebraic varieties 14N20 Configurations and arrangements of linear subspaces
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