Polynomial Lower Bounds for Arithmetic Circuits over Non-Commutative Rings

2026-04-23

AI Summary

This work investigates lower bounds on the number of multiplication gates required to compute explicit polynomials over non-commutative rings in the arithmetic circuit model. By constructing explicit n-variate degree-d polynomials and leveraging techniques from algebraic complexity theory together with structural analysis of non-commutative circuits, the authors establish the first super-linear polynomial lower bounds in this setting. Specifically, they prove that any non-commutative arithmetic circuit computing an n-variate degree-n polynomial over an arbitrary field requires at least Ω(n^{1.5}) multiplication gates, and for general degree d ≥ 2, the lower bound is Ω(d√n). These results substantially improve upon previously known bounds, which were only slightly super-linear, and constitute the first polynomial-degree lower bounds for non-commutative computation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We prove a lower bound of $Ω\left(n^{1.5}\right)$ for the number of product gates in non-commutative arithmetic circuits for an explicit $n$-variate degree-$n$ polynomial $f_{n}$ (over every field). We observe that this implies that over certain non-commutative rings $R$, any arithmetic circuit that computes the induced polynomial function $f_{n}: R^n \rightarrow R$, using the ring operations of addition and multiplication in $R$, requires at least $Ω\left(n^{1.5}\right)$ multiplications. More generally, for any $d\geq 2$ and sufficiently large $n$, we obtain a lower bound of $Ω\left(d\sqrt{n}\right)$ for $n$-variate degree-$d$ polynomials, for both these models. Prior to our work, the only known lower bounds for the size of non-commutative circuits, or for the size of arithmetic circuits over any ring, were slightly super-linear in $\max\{n,d\}$: $Ω\left(n\log d\right)$ by Baur and Strassen, and $Ω\left(d\log n\right)$ by Nisan. (Nisan's bound was proved for non-commutative arithmetic circuits and implies a bound for arithmetic circuits over non-commutative rings by our observation).

Problem

• arithmetic circuits
• non-commutative rings
• lower bounds
• polynomial complexity
• multiplication gates

Innovation

• non-commutative arithmetic circuits
• polynomial lower bounds
• arithmetic complexity
• multiplication gates
• ring computations