# 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…

## 2 Citations

Lower bounds for Polynomial Calculus with extension variables over finite fields

- Mathematics, Computer Science
- 2022

An unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over Fp with M extension variables requires size exp ( Ω(n2/(κ22κ(M+ n log(n)))) ) .

Simple Hard Instances for Low-Depth Algebraic Proofs

- Computer Science, MathematicsArXiv
- 2022

We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over ﬁelds of zero characteristic. Speciﬁcally, we show that the…

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