Felix Klein's paper on real flexes vindicated

In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein’s arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples. 0. Introduction. Recently, there has been a renewed interest for enumerative problems over the reals in algebraic geometry [1], which were abandoned after a few attempts by the founders of modern algebraic geometry. In general, in algebraic geometry, the number of solutions of an enumerative problem over the reals is bounded by the number of solutions over the complex. So, two natural questions arise: 1) Is it possible to arrange that all the solutions are real? If it is the case, the problem is called fully real in [6]. 2) Which intermediate number of solutions can be obtained?