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).

翻译：我们证明了一个显式的$n$变量、次数为$n$的多项式$f_{n}$（在任意域上）在非交换算术电路中乘积门数量具有$\Omega\left(n^{1.5}\right)$的下界。我们指出，这蕴含在特定非交换环$R$上，任何使用$R$中加法和乘法环运算来计算导出多项式函数$f_{n}: R^n \rightarrow R$的算术电路，至少需要$\Omega\left(n^{1.5}\right)$次乘法。更一般地，对于任意$d\geq 2$和足够大的$n$，我们在两种模型下均获得关于$n$变量、次数为$d$的多项式的$\Omega\left(d\sqrt{n}\right)$下界。在本文工作之前，已知的非交换电路规模下界，或任意环上算术电路规模的下界，仅略超线性于$\max\{n,d\}$：Baur和Strassen给出的$\Omega\left(n\log d\right)$，以及Nisan给出的$\Omega\left(d\log n\right)$。（Nisan的下界是针对非交换算术电路证明的，根据我们的观察，该下界蕴含非交换环上算术电路的下界。）