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. Specifically, for any nonzero polynomial $f$ in this ideal, we construct a small depth-three $f$-oracle circuit that approximates the $\Theta(r^{1/3}) \times \Theta(r^{1/3})$ determinant in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by $r \times r$ minors is at least as hard to approximately compute as the $\Theta(r^{1/3}) \times \Theta(r^{1/3})$ determinant. 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. Leveraging connections between the complexity of polynomial ideals and other questions in algebraic complexity, our results provide a generic recipe that allows lower bounds for the determinant to be applied to other problems in algebraic complexity. We give several such applications, two of which are highlighted below.

$\bullet$ We prove new lower bounds for the Ideal Proof System of Grochow and Pitassi. Specifically, we give super-polynomial lower bounds for 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. Moreover, we show that for many natural circuit classes, 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 the closure of low-depth circuits. For any $\varepsilon > 0$, we construct generators with seed length $O(n^\varepsilon)$ that hit $n$-variate low-depth circuits. Our generators 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 degree and circuit complexity of the generator.