A distance from free to dummy indices is defined. The distance is invariant with respect to both monoterm symmetries and bottom antisymmetry. Using the distance invariant, we present an index-replacement algorithm. We then develop two normalization algorithms. One is with respect to monoterm symmetries and has complexity lower than known algorithms; the other allows one to determine the equivalence of indexed polynomials in the Einstein summation ring ${\mathcal{R}}_2[\partial ̸]$.

中文翻译：

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距离不变法用于索引微分的归一化

定义了从空闲索引到虚拟索引的距离。相对于单项对称性和底部反对称性，距离是不变的。使用距离不变性，我们提出了一种索引替换算法。然后，我们开发两种归一化算法。一种是关于单项对称性，其复杂度低于已知算法。另一个允许一个确定爱因斯坦求和环中索引多项式的等价性${\mathcal{[R}}_2[\partial ̸]$。