Maximally Inflected Real Rational Curves

Maximally Inflected Real Rational Curves

We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construc...

On Maximally Inflected Hyperbolic Curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert’s method we show that for any integers d and r such that 4 ≤ r ≤ 2d − 2d, there is a non-singular hyperbolic curve of degree 2d in R with exactly r line segments in the boundary of its convex hull. We also give a complete classification of possib...

MAXIMALLY INFLECTED REAL RATIONAL CURVES VIATCHESLAV KHARLAMOV AND FRANK SOTTILE Dedicated to V. I. Arnold on the occassion of his 65th birthday

We begin the topological study of real rational plane curves all of whose inflection points are real. The existence of such curves is implied by the results of real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the ...

Real Rational Curves in Grassmanniansfrank

Real Rational Curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Similarly, given any problem of enumerating p-planes incide...