Gr (?) Bner-based Theory Of Polynomial Decomposition And Hamiltonian Cycle Problems

The principal jobs are to discuss the applies of the Grobner basis in factorization of polynomial,second-order matrix over polynomial and in finding all Hamiltonian circle in a planar graph.The article has five chapters. The first chapter is introduction. The author has introduced the influences of the computer on mathematics, principal concepts in computing algebra and computer algebra, mathem-atic software in computer algebra—Maple, and formation of the theory of Grobner basis. In Chapter 2, produced basic knowlige of Grobner basis, content algorithm of Grobner basis, improve algorithm of Grobner basis and Grobner basis on ring.In Chapter 3, we have discussed the factoring of polynomials and obtained the judging means of factoring polynomials on the basis of the theory of Grobner basis—transforming the question of polynomial factoring into the question of equational solutions and solving Grobner basis G of ideal which was formed by equational polynomials, we have known the polynomial can factor or by G , or else it can factor. In addition, we have obtained lots of characters which related to polynomials. Proved a second-order matrix over polynomial is factorable when their determinant is factorable, given a method with Grobner basis to factor it.Any Hamiltonian circle in a planar graph determines at least one dyeing solution and if there exists a Hamiltonian circle, there should exist a dyeing scheme which render the Hamiltonian circle is the boundary of the planes dyed with two colors and with another two colors is proven using equivalent relation derived from the card conception of planar graph in the Chapter V. Accordingly, an algorithm to get all the Hamiltonian circles in a planar graph is given by the combination of this property and Grobner basis. And at the end, this algorithm is realized through programming by the use of Grobner basis...