Real algebraic curves, the moment map and amoebas. (English) Zbl 1073.14555

Mikhalkin, G.

Ann. Math. (2) 151, No. 1, 309-326 (2000).

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