On the norm of products of polynomials on ultraproduct of Banach spaces

The purpose of this article is to study the problem of finding sharp lower bounds for the norm of the product of polynomials in the ultraproducts of Banach spaces $(X_i{)}_{\mathfrak{U}}$. We show that, under certain hypotheses, there is a strong relation between this problem and the same problem for the spaces $X_i$.