De-bordering is the task of proving that a border complexity measure is bounded from below, by a non-border complexity measure. This task is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's Geometric Complexity Theory (GCT) approach to settle the P \neq NP conjecture. Currently, very few de-bordering results are known. In this work, we study the question of de-bordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures, in the context of invariant theory, algebraic geometry and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only *linear* in the degree. All previous results were known to be exponential in the degree. According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result, and in this way we de-border Kumar's complexity, and obtain a new formulation of border Waring rank, up to a factor of the degree. We phrase this new formulation as the orbit closure problem of the product-plus-power polynomial, and we successfully de-border this orbit closure. We fully implement the GCT approach against the power sum, and we generalize the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the points and representation theoretic branching rules, rather than explicit multilinear computations. Furthermore, we realize that the generalization of our converse of Kumar's theorem to square matrices gives a homogeneous formulation of Ben-Or and Cleve (SICOMP'92). This results ...

翻译：摘要：去边界化是证明某个边界复杂性度量可由非边界复杂性度量从下方界定的任务。这一任务的核心在于理解Valiant的行列式与永久式猜想之间的差异，以及Mulmuley与Sohoni提出的几何复杂性理论（GCT）方法对P ≠ NP猜想的求解。目前，已知的去边界化结果极少。本文研究多项式边界Waring秩的去边界化问题。Waring秩与边界Waring秩是不变量理论、代数几何及矩阵乘法算法中广泛研究的度量。我们首次得到了一个Waring秩上界，该上界关于边界Waring秩呈指数增长，而关于次数仅呈*线性*增长。此前所有已知结果均关于次数呈指数增长。根据Kumar近期令人惊讶的结果（ToCT'20），小的边界Waring秩意味着该多项式可被近似为一个常数与少量线性多项式乘积的和。我们证明了Kumar结果的逆命题，从而实现了对Kumar复杂性的去边界化，并得到了边界Waring秩的一个新刻画（至多相差一个次数因子）。我们将这一新刻画表述为“乘积加幂”多项式的轨道闭包问题，并成功对该轨道闭包进行了去边界化。我们完全实现了针对幂和多项式的GCT方法，并将Ikenmeyer-Kandasamy（STOC'20）的思想推广至这一新轨道闭包。由此，我们得到了新的重数障碍，这些障碍仅由点的对称性和表示论的分支规则构造，而无需显式多重线性计算。此外，我们注意到将Kumar定理的逆定理推广至方阵情形，可给出Ben-Or与Cleve（SICOMP'92）的齐次形式。这导致……