Maximally inflected real rational curves

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 extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.