Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals
Robert Andrews,
Michael A. Forbes
Abstract:We show that any nonzero polynomial in the ideal generated by the r × r minors of an n × n matrix X can be used to efficiently approximate the determinant. Specifically, for any nonzero polynomial f in this ideal, we construct a small depth-three f -oracle circuit that approximates the Θ(r 1/3 ) × Θ(r 1/3 ) determinant in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by r × r minors is at least as hard to approximately … Show more
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