Grbner基与联立方程式的解法

Grbner bases and the system of equations

固定一个项序,利用Buchberger算法求多项式环S=C[x1,x2,…,xn]上的理想I的Grbner基。根据S上任意多项式f(x1,x2,…,xn)用Grbner基表示时其余项唯一的特点,将其应用到求解联立方程和求满足特定条件的多项式值等问题,从而得出Grbner基在求解多元非线性方程组方面的一个行之有效的方法,该方法为解决诸如此类数学建模问题开辟了一个新途径。

For a fixed monomial ordering, we can find the GrObner basis of ideal I on the poly- nomial ring S=C[xl ,x2 ,.. ,xn] by using the method of Buchberger algorithm. When Gr6bner basis is used to express an arbitrary polynomial f（xl ,x2,..,xn） on S, the remainders are unique. This can be applied to resolve the problems like the system of equations and the value of polynomial which satisfies specific conditions. Therefore, the results indicate Gr6bner bases is a feasible and effective way to deal with multivariate nonlinear equations, which provides a new method to solve those mathematical modeling issues.