Inflection Points of Real and Tropical Plane Curves

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.

[1]  D. Alessandrini,et al.  On the tropicalization of the Hilbert scheme , 2009, 0912.0082.

[2]  Brian Osserman,et al.  Lifting Tropical Intersections , 2010, 1007.1314.

[3]  Grigory Mikhalkin.  Decomposition into pairs-of-pants for complex algebraic hypersurfaces , 2002 .

[4]  Grigory Mikhalkin.  Tropical geometry and its applications , 2006 .

[5]  Oleg Viro,et al.  Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7 , 1984 .

[6]  Oleg Viro,et al.  Some integral calculus based on Euler characteristic , 1988 .

[7]  B. Sturmfels,et al.  First steps in tropical geometry , 2003, math/0306366.

[8]  Frank Sottile,et al.  Maximally Inflected Real Rational Curves , 2002, math/0206268.

[9]  Oleg Viro,et al.  Dequantization of Real Algebraic Geometry on Logarithmic Paper , 2000, math/0005163.

[10]  Felice Ronga.  Felix Klein's paper on real flexes vindicated , 1998 .

[11]  Kristin M. Shaw,et al.  A Tropical Intersection Product in Matroidal Fans , 2010, SIAM J. Discret. Math..

[12]  Joseph Rabinoff.  Tropical analytic geometry, Newton polygons, and tropical intersections , 2010, 1007.2665.

[13]  Rekha R. Thomas,et al.  Computing tropical varieties , 2007, J. Symb. Comput..

[14]  Felix Klein.  Eine neue Relation zwischen den Singularitäten einer algebraischen Curve , 1876 .

[15]  R. Bieri,et al.  The geometry of the set of characters iduced by valuations. , 1984 .

[16]  On the number of connected components of the parabolic curve , 2009, 0904.4652.

[17]  R. Fricke,et al.  Über den Verlauf der Abelschen Integrale bei den Kurven vierten Grades , 1922 .

[18]  O. Viro,et al.  Progress in the topology of real algebraic varieties over the last six years , 1986 .

[19]  I. Shafarevich.  Basic algebraic geometry , 1974 .