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

TL;DR

This paper introduces new uniform upper bounds for symmetric bilinear complexity in finite field extensions and examines Shimura curves that challenge existing bounds, advancing understanding of tensor rank limitations.

Contribution

It provides the first non-asymptotic uniform bounds for symmetric tensor rank and analyzes Shimura curves that question previous assumptions.

Findings

New uniform upper bounds for symmetric bilinear complexity

Examples of Shimura curves that do not descend over their field of moduli

Discussion on the validity of existing bounds

Abstract

We establish new upper bounds about symmetric bilinear complexity in any extension of finite fields. Note that these bounds are not asymptotical but uniform. Moreover we give examples of Shimura curves that do not descend over their field of moduli, which discusses the validity of certain published bounds.