Lower Bounds for Noncommutative Circuits with Low Syntactic Degree

Proving lower bounds on the size of noncommutative arithmetic circuits is an important problem in arithmetic circuit complexity. For explicit n variate polynomials of degree Θ(n), the best known general bound is Ω(n log n) [Strassen, 1973; Walter Baur and Volker Strassen, 1983]. Recent work of Chatterjee and Hrubeš [Chatterjee and Hrubeš, 2023] has provided stronger (Ω(n²)) bounds for the restricted class of homogeneous circuits. The present paper extends these results to a broader class of circuits by using syntactic degree as a complexity measure. The syntactic degree of a circuit is a well known parameter which measures the extent to which high degree computation is used in the circuit. A homogeneous circuit computing a degree d polynomial can be assumed, without loss of generality, to have syntactic degree exactly equal to d [Fournier et al., 2024]. We generalize this by considering circuits that are not necessarily homogeneous but have low syntactic degree. Specifically, for an explicit n variate, degree n polynomial f we show that any circuit with syntactic degree O(n) computing f must have size Ω(n¹⁺ᶜ) for some constant c > 0. We also show that any circuit with syntactic degree o(nlog n) computing the same f must have size ω(nlog n). We further analyze the circuit size required to compute f based on the number of distinct syntactic degrees appearing in the circuit. Our analysis yields an ω(nlog n) size lower bound for all but a narrow parameter regime where an improved bound is not obtained. Finally, we observe that low syntactic degree circuits are more powerful than homogeneous circuits in a fine grained sense: there exists an n variate, degree Θ(n) polynomial that has a circuit of size O(nlog ²n) and syntactic degree O(n) but any homogeneous circuit computing it requires size Ω(n²).