Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ideals, Determinants, and Straightening: Proving and Using Lower Bounds
  for Polynomial Ideals   We show that any nonzero polynomial in the ideal generated by the $r \times
r$ minors of an $n \times n$ matrix $X$ can be used to efficiently approximate
the determinant. For any nonzero polynomial $f$ in this ideal, we construct a
small depth-three $f$-oracle circuit that approximates the determinant of size
$\Theta(r^{1/3})$ in the sense of border complexity. For many classes of
algebraic circuits, this implies that every nonzero polynomial in the ideal
generated by $r \times r$ minors is at least as hard to approximately compute
as the determinant of size $\Theta(r^{1/3})$. We also prove an analogous result
for the Pfaffian of a $2n \times 2n$ skew-symmetric matrix and the ideal
generated by Pfaffians of $2r \times 2r$ principal submatrices.
  This answers a recent question of Grochow about complexity in polynomial
ideals in the setting of border complexity. We give several applications of our
result, two of which are highlighted below.
  $\bullet$ We prove super-polynomial lower bounds for Ideal Proof System
refutations computed by low-depth circuits. This extends the recent
breakthrough low-depth circuit lower bounds of Limaye, Srinivasan, and Tavenas
to the setting of proof complexity. For many natural circuit classes, we show
that the approximative proof complexity of our hard instance is governed by the
approximative circuit complexity of the determinant.
  $\bullet$ We construct new hitting set generators for polynomial-size
low-depth circuits. For any $\varepsilon > 0$, we construct generators with
seed length $O(n^\varepsilon)$ that attain a near-optimal tradeoff between
their seed length and degree, and are computable by low-depth circuits of
near-linear size (with respect to the size of their output). This matches the
seed length of the generators recently obtained by Limaye, Srinivasan, and
Tavenas, but improves on the generator's degree and circuit complexity.

31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom