On some bounds for symmetric tensor rank of multiplication in finite fields

The aim of this paper is twofold. On the one hand, we establish new upper bounds for the symmetric multiplication tensor in any extension of finite fields. Note that these bounds are not asymptotic but uniform. On the other hand, we clarify the current state of the art by giving the detailed proof of some known unpublished uniform bounds, and we discuss the validity of some current asymptotic bounds and their relation with the fields of definition of certain Shimura curves.