Ideals, determinants, and straightening: proving and using lower bounds for polynomial ideals
Robert Andrews, Michael A. Forbes · 2022
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(r1/3) × Θ(r1/3) 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 compute as the Θ(r1/3) × Θ(r1/3) determinant. We also prove an analogous result for the Pfaffian of a 2n × 2n skew-symmetric matrix and the ideal generated by Pfaffians of 2r × 2r principal submatrices.