# 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={Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing},
year={2021}
}```
• Published 1 December 2021
• Mathematics, Computer Science
• Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
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…
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