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<p>Riemman metric tensor (Rmt) plays a significant role in deducing basic formulas and equations arising in differential geometry and (pseudo-) Riemannian manifolds. It is a fundamental and challenging problem to determine the equivalence of indexed differential Rmt polynomials. This paper solves the problem by extending Gröbner basis theory and the previous work on the computational theory for indexed differentials. <i>L</i>-expansion of an indexed differential Rmt polynomial is defined. Then a decomposed form of the Gröbner basis of defining syzygies of the polynomial ring is presented, based on a partition of elementary indexed monomials. Meanwhile, the upper bound of the dummy index numbers of sim-monomials of the elements in each disjoint elementary indexed monomial subset is found. Finally, a DST-fundamental restricted ring is constructed, and the canonical form of a polynomial is confirmed to be the normal form with respect to the Gröbner basis in the DST-fundamental restricted ring.</p>

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Restricted Gröbner Basis Theory for Normalization of Indexed Differential Riemann Metric Tensor Polynomials

  • 期刊论文

  • 发表日期：

    2025年1月30日

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  • Jiang Liu,
  • Tao Wang,
  • Pingjing Hou,
  • Feng Ni,
  • Kun Zhu,
  • Leyi Zhang

摘要

Riemman metric tensor (Rmt) plays a significant role in deducing basic formulas and equations arising in differential geometry and (pseudo-) Riemannian manifolds. It is a fundamental and challenging problem to determine the equivalence of indexed differential Rmt polynomials. This paper solves the problem by extending Gröbner basis theory and the previous work on the computational theory for indexed differentials. L-expansion of an indexed differential Rmt polynomial is defined. Then a decomposed form of the Gröbner basis of defining syzygies of the polynomial ring is presented, based on a partition of elementary indexed monomials. Meanwhile, the upper bound of the dummy index numbers of sim-monomials of the elements in each disjoint elementary indexed monomial subset is found. Finally, a DST-fundamental restricted ring is constructed, and the canonical form of a polynomial is confirmed to be the normal form with respect to the Gröbner basis in the DST-fundamental restricted ring.