# 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} }

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…

## 5 Citations

### Simple Hard Instances for Low-Depth Algebraic Proofs

- Computer Science, Mathematics2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 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…

### No short polynomials vanish on bounded rank matrices

- MathematicsBulletin of the London Mathematical Society
- 2023

We show that the shortest nonzero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that…

### Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits

- Mathematics2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

Mulmuley and Sohoni (2001) proposed an ambitious program, the Geometric Complexity Theory (GCT), to prove $P\neq NP$ and related conjectures using algebraic geometry and representation theory.…

### On Matrix Multiplication and Polynomial Identity Testing

- Mathematics2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting…

### Lower bounds for Polynomial Calculus with extension variables over finite fields

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 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)))) ) .

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