RESEARCH ANNOUNCEMENTS——The Construction of Eigenvalue Problem Equivalent to Multivariate Polynomial System and the Groebner Basis

In this paper we will show that one cau build up a joint eigenvalue problem eq-uivalent to the. given system. By this way, finding the solutions of the given systemis equivalent to finding all eigenvalues and eigenvectors of one matrix or matrix pen-cil. For the special case that the system has finite isolated solutions, we can obtainall solutions through computing the eigenvalues and eigenvectors of a matrix whichcan Le obtained by Gauss-Jordan elimination. Furthermore, we also find that one canget Groebner Basis for the ideal geuerated by the given system iu this way. For any polynomial f(x)∈K[x_1,x_2,…,x_n],f(x) can be written as     

Second Equivalence Theorem of Multivariate Polynomial System and Eigenproblem

In[1,2] we have constructed the Joint eigenproblem equivalent to multivariatepolynomial system and presented an algorithm to solve latter.In this paper we willstudy another equivalence theorem for the same problem.The computation of thecorresponding eigenvalues is numerically stable. Consider multivarite polynomial system