Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits. Debordering is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's variation which uses border determinantal complexity. The debordering of matrix multiplication tensors by Bini played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Currently, very few debordering results are known. In this work, we study the question of debordering 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 known results were exponential in the degree. For polynomials with constant border Waring rank, our results imply an upper bound on the Waring rank linear in degree, which previously was only known for polynomials with border Waring rank at most 5.

翻译：边界复杂度通过极限（或拓扑闭包）定义，使得任何可被低复杂度函数任意逼近的函数本身具有低边界复杂度。去边界化是指基于边界复杂度度量，证明某个非边界复杂度度量的上界，从而消除极限的过程。去边界化是理解Valiant行列式与积和式猜想、以及Mulmuley与Sohoni基于边界行列式复杂度的变体之间差异的核心。Bini对矩阵乘法张量的去边界化在高效矩阵乘法算法的发展中起到了关键作用。因此，去边界化既在建立计算复杂度下界方面得到应用，也促进了算法设计。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领域中备受关注的概念。我们首次证明了Waring秩的上界：该上界相对于边界Waring秩呈指数增长，且仅关于次数呈线性增长。而此前所有已知结果的上界均关于次数呈指数增长。对于边界Waring秩为常数的多项式，我们的结果给出了关于次数呈线性增长的Waring秩上界，这类结果此前仅对边界Waring秩不超过5的多项式成立。