On the homogeneous Gröbner basis for tensors

Algorithmic methods of commutative algebra based on the involutive and Gröbner bases technique are efficient means for completion of equations governing dynamical systems to involution. At the same time, when working with high-dimensional tensor quantities, direct use of standard functions for calculating Gröbner bases, which are built in computer algebra systems Maple and Mathematica , requires much memory. However, being multilinear forms, tensors admit special grading that makes it possible to classify polynomials in terms of their degree of homogeneity. With regard to this feature, we propose to use a special homogeneous Gröbner basis, which allows us to avoid difficulties associated with large amount of computation. Such a basis is constructed step by step, as the degree of the polynomial grows. As an example, an algorithm for constructing the homogeneous basis in a finite-dimensional Hamiltonian system with many polynomial constraints (the so-called Yang-Mills mechanics) is presented.