Border Complexity of Symbolic Determinant under Rank One Restriction
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VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_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"aser, 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 + ∑_i=1^n A_ix_i where for each 1≤ i ≤ n, A_i is of rank one. It has been studied extensively [Edmonds(1968), Edmonds(1979)] and efficient identity testing algorithms are known [Lov"asz (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 Plücker embedding of) a Grassmannian variety is closed.
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