On the multilinear Bohnenblust--Hille constants: complex versus real case

The results of this note arise a rupture between the behavior of the real and complex best known constants for the multilinear Bohnenblust--Hille inequality; in one side, for real scalars, we show that new upper bounds for the real Bohnenblust--Hille inequality (the best up to now) can be obtained via a somewhat "chaotic" combinatorial approach, while in the complex case the combinatorial approach giving the best known constants seems to be fully controlled. We believe that the understanding of this fact is a challenging problem that may shed some new light to the subject. As a byproduct of our results we present new estimates for the constants of the Bohnenblust--Hille inequality as well as new closed formulas.