多项式的化简和标准型是计算机代数中的一个关键而基本的古老研究课题，然而指标多项式满足Einstein求和约定，其本质上是采用符号表示乘法与加法的复合，从而引起有关的代数结构发生根本变化。关于该代数结构的Groebner基等理论尚不存在，这成为计算机代数中一个挑战性问题。另一方面，现代微分几何和物理学中常常涉及指标多项式的大量复杂、技巧性计算和推导，迫切需要关于指标多项式化简和标准型的理论与算法。而现有算法效率较低或只面向特定指标多项式而不可推广。针对上述问题，本项目首先通过构造最小“限制”幺半群环，进而运用Groebner基理论建立指标多项式关于内部对称性标准型，以及微分指标多项式标准型一般性算法，其次提炼多层次距离不变量，发展基于距离不变量的指标赋值法，用于建立高效标准型算法。最后应用于微分几何中张量判定、坐标变换下的规律问题的自动求解，和关于多种重要而基本张量的公式及定理的自动证明。

The simplification and normalization of algebraic expressions is an "ancient" research topic, which is both important and fundamental in computer algebra. However, indexed expressions obey Einstein summation convention which stands for a composition of multiplication and addition, making the algebraic structure of indexed expressions much more complex. And Groebner basis theory is not yet powerful enough to deal with such algebraic structure. Therefore, the simplification and normalization of indexed expressions remains to be a challenging problem. On the other hand, modern differential geometry and physics often involve massive manual calculations of indexed expressions which are tricky and cumbersome to perform. They call for both theoretical results and algorithms for simplifying and normalizing indexed polynomials. However, existing algorithms either have low efficiency or cannot be extended. To solve these problems, this project first develops a general normalization algorithm of indexed polynomials with respect to interior symmetries, and a normalization algorithm of indexed differentials, by extending Groebner basis theory due to the method of constructing minimal restricted monoid rings. Secondly, provides an index replacement algorithm based on multi-level distance invariants, which is then used to establish efficient normalization algorithms. Finally the algorithms are applied to mechanically solving tensor verification problem and the problem of finding transformation rules of indexed functions under coordinate transformation, and to mechanically proving and exploring formulas and theorems involving several important and fundamental tensors in differential geometry.