The Entry Formula For A Mixed Resultant Matrix Of N-variable Polynomial System

This paper studies the entry formula for a mixed resultant matrix of double and three–variable polynomial system.Mainly to solve: with starting from the definition of the mixed resultant matrix of double-variable polynomial system,transforming polynomial division step to the expression of entry formula,and using the mathematical induction method to find the expression formula of the Dixon quotient,which make us restrict the matrix according to the block structure of mixed resultant matrix.It draws the conclusion we want.Transform double-variable polynomial system to three-variable polynomial system,and study the entry formula of the mixed resultant matrix in the same way,the results are similar.This not only avoid the process of polynomial division,but also calculate all of the 2 order determinant with the help of computer,and directly substitute into the proposed formula,which greatly simplify the calculation process.For polynomial systems with more variables,it is bound to bring about an increase in the complexity of the expression,the introduction of new tools will be inevitable,here is not to discuss.First,it introduces the background and significance of the resultant method and summarize the development history and the present research situation of resultant method.Then it gives some basic concepts and the definition of mixed resultant matrix of two-variable polynomial system-----offer the method to construct of mixed resultant matrix of two-variable polynomial system.For the mixed resultant matrix of three-variable polynomial system,the definition method is similar.Thirdly,it introduces corresponding simplify symbols to express our main conclusion----the structure of the mixed resultant matrix,and prove by mathematical induction.Last,it illustrates the effectiveness and practicability of this formula through three concrete examples so that we can have a better understanding of the entry formula of mixed resultant.All these play an important role in studying the deep structure of mixed resultant matrix.