On the typical rank of real bivariate polynomials

Here we study the typical rank for real bivariate homogeneous polynomials of degree d⩾6 (the case d⩽5 being settled by P. Comon and G. Ottaviani). We prove that d−1 is a typical rank and that if d is odd, then (d+3)/2 is a typical rank. The Comon-Ottaviani conjecture was later completely solved by G. Blekherman.