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 possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree 6.