On the real polynomial Bohnenblust–Hille inequality

It was recently proved by F. Bayart and the fourth and fifth authors that the complex polynomial Bohnenblust–Hille inequality is subexponential. Here, we show that (for real scalars) this no longer holds. Moreover, we show (among other results) that, if DR,mDR,m stands for the real Bohnenblust–Hille constant for m-homogeneous polynomials, thenlim⁡supm⁡DR,m1/m=2, a quite surprising result having in mind that the exact value of the Bohnenblust–Hille constants is still a mystery.