A note on the polynomial Bohnenblust–Hille inequality

Recently, in a paper published in the Annals of Mathematics, it was shown that the Bohnenblust–Hille inequality for (complex) homogeneous polynomials is hypercontractive. However, and to the best of our knowledge, there is no result providing (nontrivial) lower bounds for the optimal constants for n -homogeneous polynomials ( n > 2 ). In this short note we provide lower bounds for these famous constants.