VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ where the size of each $A_i$ is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation. The power of approximation is well understood for some restricted models of computation, e.g., the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Bl"{a}ser, Ikenmeyer, Mahajan, Pandey, and Saurabh (2020)], whereas the approximative closure of the last one captures the whole class of polynomial families computable by polynomial-sized formulas [Bringmann, Ikenmeyer, and Zuiddam (2017)]. In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ where for each $1\leq i \leq n$, $A_i$ is of rank one. It has been studied extensively [Edmonds(1968), Edmonds(1979)] and efficient identity testing algorithms are known [Lov"{a}sz (1989), Gurjar and Thierauf (2020)]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Pl\"{u}cker embedding of) a Grassmannian variety is closed.

翻译：[translated abstract in Chinese] VBP 是指可由形如 $A_0 + \sum_{i=1}^n A_ix_i$ 的符号矩阵的行列式计算的多元多项式族类，其中每个 $A_i$ 的尺寸关于变量个数呈多项式级增长（等价地，可由多项式规模代数分支程序（ABP）计算）。几何复杂度理论（GCT）中的一个重大开放问题是判断 VBP 在近似下是否封闭。对于某些受限计算模型，近似的威力已被充分理解，例如深度为二的电路类、只读一次无意识 ABP（ROABP）、单调 ABP、有界顶部扇入深度为三的电路以及宽度为二的 ABP。前三类已知在近似下封闭 [Bl"{a}ser, Ikenmeyer, Mahajan, Pandey, and Saurabh (2020)]，而最后一类的近似闭包则囊括了可由多项式规模公式计算的全部多元多项式族 [Bringmann, Ikenmeyer, and Zuiddam (2017)]。本文考虑 VBP 的一个子类：由形如 $A_0 + \sum_{i=1}^n A_ix_i$ 的符号矩阵的行列式计算，其中每个 $1\leq i \leq n$ 对应的 $A_i$ 均为秩一矩阵。该子类已被广泛研究 [Edmonds(1968), Edmonds(1979)]，且已知其高效的恒等测试算法 [Lov"{a}sz (1989), Gurjar and Thierauf (2020)]。我们证明该子类在近似下封闭。用代数几何的语言表述，我们证明由格拉斯曼簇（其 Plücker 嵌入）中成对点的坐标积所构成的集合是封闭的。