On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

In this paper, we give a survey of the known results concerning the tensor rank of multiplication in finite extensions of finite fields, enriched with some unpublished recent results, and we analyze these to enhance the qualitative understanding of the research area. In particular, we identify and clarify certain partially proved results and emphasise links with open problems in number theory, algebraic geometry, and coding theory. Bibliography: 92 titles.