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- W3217497773 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(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." @default.
- W3217497773 created "2021-12-06" @default.
- W3217497773 creator A5004819634 @default.
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- W3217497773 date "2022-06-09" @default.
- W3217497773 modified "2023-10-14" @default.
- W3217497773 title "Ideals, determinants, and straightening: proving and using lower bounds for polynomial ideals" @default.
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- W3217497773 doi "https://doi.org/10.1145/3519935.3520025" @default.
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