Typical Ranks of ternary cubic forms over R

In this thesis we consider the problem of the classification of real ternary cubics, that is, plane cubic curves with real coefficients, with respect to an arithmetic invariant, the rank, and we give the decomposition of each real ternary cubic form. We prove a theorem that characterizes the reducible cubic which factors as a product of imaginary conic and a real line with respect to rank and this is a new result in the theory of real plane cubic curves.