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De-bordering is the task of proving that a border complexity measure isbounded from below, by a non-border complexity measure. This task is at theheart of understanding the difference between Valiant's determinant vspermanent conjecture, and Mulmuley and Sohoni's Geometric Complexity Theory(GCT) approach to settle the P \neq NP conjecture. Currently, very fewde-bordering results are known.  In this work, we study the question of de-bordering the border Waring rank ofpolynomials. Waring and border Waring rank are very well studied measures, inthe context of invariant theory, algebraic geometry and matrix multiplicationalgorithms. For the first time, we obtain a Waring rank upper bound that isexponential in the border Waring rank and only *linear* in the degree. Allprevious results were known to be exponential in the degree.  According to Kumar's recent surprising result (ToCT'20), a small borderWaring rank implies that the polynomial can be approximated as a sum of aconstant and a small product of linear polynomials. We prove the converse ofKumar's result, and in this way we de-border Kumar's complexity, and obtain anew formulation of border Waring rank, up to a factor of the degree. We phrasethis new formulation as the orbit closure problem of the product-plus-powerpolynomial, and we successfully de-border this orbit closure. We fullyimplement the GCT approach against the power sum, and we generalize the ideasof Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, weobtain new multiplicity obstructions that are constructed from just thesymmetries of the points and representation theoretic branching rules, ratherthan explicit multilinear computations.  Furthermore, we realize that the generalization of our converse of Kumar'stheorem to square matrices gives a homogeneous formulation of Ben-Or and Cleve(SICOMP'92). This results ...

De-bordering and Geometric Complexity Theory for Waring rank and related models