On real typical ranks

We study typical ranks with respect to a real variety X. Examples of such are tensor rank (X is the Segre variety) and symmetric tensor rank (X is the Veronese variety). We show that any rank between the minimal typical rank and the maximal typical rank is also typical. We investigate typical ranks of n-variate symmetric tensors of order d, or equivalently homogeneous polynomials of degree d in n variables, for small values of n and d. We show that 4 is the unique typical rank of real ternary cubics, and quaternary cubics have typical ranks 5 and 6 only. For ternary quartics we show that 6 and 7 are typical ranks and that all typical ranks are between 6 and 8. For ternary quintics we show that the typical ranks are between 7 and 13.