Resumo:
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.
