• Corpus ID: 244799306

Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals

@article{Andrews2021IdealsDA,
  title={Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals},
  author={Robert Andrews and Michael A. Forbes},
  journal={ArXiv},
  year={2021},
  volume={abs/2112.00792}
}
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… 
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