The Gröbner Bases Algorithm and its Application in Polynomial Ideal Theory

Due to the huge computations in the mathematics and engineering, it is necessary to reduce the computational complexity in advance. A Gröbner bases is a set of multivariate nonlinear polynomials, which can treat a unit of many problems as a basis, and then take advantage of these bases to simplify the objective polynomial such that a simple multivariate polynomial system can be obtained. This paper first reviews the original algorithm of Gröbner bases and the improvement of the original algorithm by N. K. Bose, then gives two applications of Gröbner bases algorithm to the practical problems. The results show that Gröbner bases algorithm can greatly reduce the computation complexity and improve the execution efficiency to some extent.