Sufficient conditions for Strassen's additivity conjecture

TL;DR

This paper establishes sufficient conditions under which Strassen's additivity conjecture holds for Waring and cactus ranks, providing criteria for when ranks and decompositions are additive for sums of forms.

Contribution

It introduces new sufficient conditions for the additivity of Waring and cactus ranks, advancing understanding of rank decompositions in polynomial algebra.

Findings

Waring rank additivity holds under specific conditions

Additivity of cactus ranks is characterized by new criteria

Decompositions of sums can be expressed as sums of individual decompositions

Abstract

We give a sufficient condition for the strong symmetric version of Strassen's additivity conjecture: the Waring rank of a sum of forms in independent variables is the sum of their ranks, and every Waring decomposition of the sum is a sum of decompositions of the summands. We give additional sufficient criteria for the additivity of Waring ranks and a sufficient criterion for additivity of cactus ranks and decompositions.