Aditividade do posto de tensores pequenos de três fatores

Abstract

This work explores the application of advanced concepts of algebra and algebraic geometry to understand some attempts to solve the so-called Strassen Conjecture, which consists of considering the union of two bilinear systems, each depending on different variables, and determining whether the multiplicative complexity of this union is equal to the sum of the multiplicative complexities of both systems. Our study relates these bilinear systems and their multiplicative complexities to tensor spaces and their ranks, respectively. We will restrict our study to the case of three-factor tensor spaces, developing the theoretical knowledge needed to support the current conclusions and establish new research directions. It is common knowledge that the Conjecture is not true, however, we will study some special cases in which the Conjecture holds, using concepts and results relating to projective spaces, linear transformations and their properties.