论文标题
产品加功率的几何复杂性理论
Geometric complexity theory for product-plus-power
论文作者
论文摘要
根据库马尔最近的惊人结果（TOCT'20），一个小的边界警告等级意味着多项式可以作为线性多项式的常数和小型产物的总和。我们证明了库马尔结果的相反，并在库马尔的结果中建立了边界警告等级与计算模型之间建立紧密的联系。通过这种方式，我们获得了边境警告等级的新表述，最多是该学位的一倍。我们将这种新公式连接到产品加功率多项式的轨道闭合问题。我们从两个方向研究了这个轨道封闭：1。我们辩论此轨道封闭和一些相关的轨道封闭，即证明轨道闭合中的所有点都具有小的非生物代数分支程序。 2。我们通过将Ikenmeyer-Kandasamy（STOC'20）的思想推广到这种新的轨道封闭中，完全实施了对权力总和的几何复杂性理论方法。这样，我们获得了仅由多项式的对称性构建的新多重性障碍物。
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 establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree. We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions: 1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs. 2. We fully implement the geometric complexity theory approach against the power sum by generalizing 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 polynomials.
