Research Of Gr(?)bner Bases, Reducing And For Their Application

The article have five chapters, the first chapter is introduction. Chapter 2 give an account of the reducing of polynomial group, adopting a kind of changed sequence to drop the degree of polynomial, the 3th chapter tell about the improved arithmetic of Grobner basis. It is a new way based on arithmetic of GrobnerNew by adopting the locally analytic method. The process of this new way is that categorizes the leading term of generator of the polynomial in the idea according to related expressions firstly, then analysis every categories. If a polynomial can be reduced to a remainder polynomial by a quotient polynomial in the idea, then it can be replaced by the remainder polynomial, the new way can withhold the number of middle term increasing too fast and the degree of polynomial being too high.Chapter 4 tell about the algebraic varieties of ideal quotients in the pre-prime ideal, if Q is P-pre-prime idea of polynomial rings k[x1, x2....,xn], and J is the subset of rings, assumed , then the algebraic varieties of idea quotient V(Q:J)= . Assumed then is the root idea of Q:J). Assumed then The last chapter is about the application of Grobner basis. First, applying the map of polynomials and Grobner basis to solve the direct expression of the equation of parameters, then applying the Grobner basis and reducing to research the optimal route of two joint point in graphic theory. The first step, a pair of adjacent joint point can be expressed by a polynomial form, example, point jc/ and Xj , they have a route reach directly , the length of route being a,y , can be expressed as Xj-Xj-dij, then we classify all the polynomial according the leading term , form a polynomial table F , its depth is 2. If there are optimal route in the graphic, then the Grobner basis of generating idea by F is {!}, Assumed what we research is the optimal route of x* to xm in the graphic,the method is using polynomial xm-xk reducing the unit of table F. At last, we can get a constant being the length of the route, For there are lots of different route. We can get a group constant, the minimal constant is half of the shortest routed, that is what we are research.