Real Rank of Tensors

Some real tensors have real rank strictly bigger than the complex rank. This phenomenon, that you will never meet by playing with matrices, is not pathological but actually happens on tensor sets of positive measure. Playing with 2*2*2 tensors, you will meet this with probability (1-pi/4)=0.21.Few general facts are known on real rank, although Sylvester proved in 1864 the nontrivial fact that the real rank of a univariate polynomial is greater or equal than the number of its real roots. We will report on some recent results by Comon, Blekherman, Teitler and others,including Banchi's classification of real rank for bivariate cubic polynomials. We conclude with some open questions.