We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characterisitic zero.

中文翻译：

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对称算术电路

我们介绍对称算术电路，即具有自然对称性限制的算术电路。在计算定义在变量矩阵（例如行列式或永久式）上的多项式的电路的上下文中，限制相当于要求电路的形状在矩阵的行列排列下保持不变。我们为任何对称电路的大小建立了无条件的、接近指数的、下界，用于计算除 2 以外的任何特征域上的永久。相比之下，我们表明存在多项式大小的对称电路用于计算特征域上的行列式零。