Some cases on Strassen additive conjecture

Abstract

In this article, we verify the additivity for rank of a sum of coprime monomials and bivariate polynomials generalizing the result in ([CCG12]). We also show similar results hold for cactus rank.

Geometric lower bounds for generalized ranks

“…, x n ) + y d 1 + • • • + y d s ) = r(F ) + s and r(F (x 1 , x 2 ) + G(y 1 , y 2 )) = r(F ) + r(G) (a conjecture of Strassen asserts that this should hold for all F and G involving any number of variables). See also [Woo14]. Together with the previously mentioned quadratic and binary forms, ternary cubics, and general forms, this is, as far as I know, a complete list of all forms whose Waring ranks have been determined.…”

“…Another approach to lower bounds for Waring rank itself comes from the technique used to find Waring ranks of monomials and sums of pairwise coprime monomials in [CCG12]. Very recently this method has been developed further in [CCC14] and [Woo14]. So far it has been used to find Waring ranks of increasingly general examples, namely, sums of polynomials in small numbers of linearly independent variables.…”