Algorithmic Thresholds for Refuting Random Polynomial Systems

Abstract: Consider a system of $m$ polynomial equations ${p_i(x) = b_i}_{i \leq m}$ of degree $D\geq 2$ in $n$-dimensional variable $x \in \mathbb{R}<sup>n$ such that each coefficient of every $p_i$ and $b_i$s are chosen at random and independently from some continuous distribution. We study the basic question of determining the smallest $m$ -- the algorithmic threshold -- for which efficient algorithms can find refutations (i.e. certificates of unsatisfiability) for such systems. This setting generalizes problems such as refuting random SAT instances, low-rank matrix sensing and certifying pseudo-randomness of Goldreich's candidate generators and generalizations. We show that for every $d \in \mathbb{N}$, the $(n+m)<sup>{O(d)}$-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever $m \geq O(n) \cdot (\frac{n}{d})<sup>{D-1}$. We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all $d$. We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-$4$ sum-of-squares relaxation for refuting random quadratic polynomials. Together, our results provide evidence for an algorithmic threshold for the problem at $$m \gtrsim \widetilde{O}(n) \cdot n<sup>{(1-\delta)(D-1)}$$ for $2<sup>{n<sup>{\delta}}$-time algorithms for all $\delta$.