Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field $${{\mathbb{F}}}$$F of characteristic zero. We resolve this problem by proving a $${N^{\Omega(\frac{d}{\tau})}}$$NΩ(dτ) lower bound for (nonhomogeneous) depth-three arithmetic circuits with bottom fanin at most $${\tau}$$τ computing an explicit $${N}$$N-variate polynomial of degree $${d}$$d over $${{\mathbb{F}}}$$F. Meanwhile, Nisan & Wigderson (Comp Complex 6(3):217–234, 1997) had posed the problem of proving super-polynomial lower bounds for homogeneous depth-five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of $${N^{\Omega(\sqrt{d})}}$$NΩ(d) for homogeneous depth-five circuits (resp. also for depth-three circuits) with bottom fanin at most $${N^{\mu}}$$Nμ, for any fixed$${\mu < 1}$$μ<1. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth-five circuit has bottom fanin at most $${N}$$N).

中文翻译：

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具有小底扇入的深度三算术电路的下界

Shpilka & Wigderson（IEEE 计算复杂性会议，第 87 卷，1999 年）提出了证明（非齐次）深度三算术电路的指数下界的问题，在一个域 $${{\mathbb{F}} }$$F 特征零。我们通过证明一个 $${N^{\Omega(\frac{d}{\tau})}}$$NΩ(dτ) 下界来解决这个问题，用于（非齐次）深度三算术电路，最多具有底部扇入$${\tau}$$τ 在 $${{\mathbb{F}}}$F 上计算显式的 $${N}$$N 次多项式 $${d}$$d。与此同时，Nisan & Wigderson (Comp Complex 6(3):217–234, 1997) 提出了证明同构深度五运算电路的超多项式下界的问题。在特征为零的域上，我们显示了 $${N^{\Omega(\sqrt{d})}}$$NΩ(d) 的下限，用于同构深度五电路（也适用于深度三电路），底部扇入最多$${N^{\mu}}$$Nμ，对于任何固定的 $${\mu < 1}$$μ<1。这部分解决了 Nisan 和 Wigderson 提出的问题，因为增加了对底部扇入的限制（一般的同质深度 5 电路的底部扇入最多为 $${N}$$N）。