Algorithmic Thresholds for Refuting Random Polynomial Systems
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Consider a system of m polynomial equations {p_i(x) = b_i}_i ≤ m of degree D≥ 2 in n-dimensional variable x ∈ℝ^n such that each coefficient of every p_i and b_is 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 ∈ℕ, the (n+m)^O(d)-time canonical sum-of-squares (SoS) relaxation refutes such a system with high probability whenever m ≥ O(n) · (n/d)^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 ≳O(n) · n^(1-δ)(D-1) for 2^n^δ-time algorithms for all δ.
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