Irreversibility of structure tensors of modules

Abstract

Determining the matrix multiplication exponent $$\omega $$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $$\omega = 2$$ by starting with structure tensors of modules of fixed degree and using arbitrary restrictions. It implies that the same is impossible by starting with $$1_A$$ -generic non-diagonal tensors of fixed size with minimal border rank. This generalizes the work of Bläser and Lysikov (Slice rank of block tensors and irreversibility of structure tensors of algebras 170, 2020). Our methods come from both commutative algebra and complexity theory.