The coamoeba of any complex algebraic plane curve V is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in (C∗)2 is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in (C∗)2, the complex algebraic plane curves whose coamoebas are of maximal area (counted with mul...

A nonsingular real algebraic plane projective curve is called a dividing curve or a curve of type I if its real point set divides its complex point set. Rokhlin’s formula, which holds for such curves, is an important step in order to classify nonsingular real algebraic plane projective curves. It gives prohibitions on the complex orientations of a curve of type I and also on its real scheme. Th...

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in R. Specifically, we show that an arrangement of n algebraic curves determines at most Cǫn 4/3+3ǫ two-rich points, provided at most n curves lie in any low degree hypersurface and at most n curves lie in any low degree surface. This result follows from a structure theorem about arra...

We show how one may sometimes perform singular ambient surgery on the complex locus of a real algebraic curve and obtain what we call a floppy curve. A floppy curve is a certain kind of singular surface in CP (2), more general than the complex locus of a real nodal curve. We derive analogs for floppy curves of known restrictions on real nodal curves. In particular we derive analogs of Fiedler’s...

There exists a duality between elliptic curves and noncommutative tori, i.e. C∗-algebras generated by the unitary operators u and v such that vu = euv. We show that this duality can be included into a general picture involving the algebraic curves of higher genus. In this way we prove that a big part of geometry of complex algebraic curves can be developed from the K-theory of a noncommutative ...

In this paper we provide an algorithm to find explicitly rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its invariant algebraic curves. The method is based on the rational parametrization of the rational invariant algebraic curves and intensively using of linear fractional transformations between two proper rational parametrizations of the same...

We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. ...

We use C-algebras to study complex algebraic curves. Our approach is based on the representation of an algebraic curve of genus g by the interval exchange transformation due to H. Masur, W. Veech et al. We study the C-algebra Oλ connected to such transformation. The main result says that the algebra Oλ, taken up to Morita equivalence, defines the curve C, up to conformal equivalence. The first ...