The Research On The Equivalence Of Several Classes Of Polynomial Matrices And Their Smith Forms

Since multi-dimensional systems are often described by polynomial matrices.The equivalence problem of multi-dimensional matrices is to simplify the formal structure of multi-dimensional systems in order to simplify the corresponding system operations.Therefore,scholars use the equivalent research on polynomial matrices to reduce the corresponding multi-dimensional matrix.system.The equivalence problem of polynomial matrix and its Smith form is a very important class of equivalence problems of polynomial matrix,but there is no universally applicable method to judge whether a general polynomial matrix is equivalent to its Smith form.Combining the existing results,this paper discusses the equivalence of two kinds of special n-D(n≥2)polynomial matrice and its Smith form.First,for a class of bivariate polynomial matrix F(z1,z2)∈ Kl×l[z1,z2],its determinant is hr(z1)(where h(z1)is an irreducible monovariate polynomial in K[z1]).By considering the bivariate polynomial of F(z1,z2)as a polynomial about the unknown element z2,that is,the term containing the unknown element z1 is regarded as a coefficient,and using the Euclidean division property on the coefficient ring,it is proved that when F(z1,z2)satisfies h(z1)|dl-r+1(F),F(z1,z2)is always equivalent to its Smith form(Il-r h(z1)·Ir)).Then the result is generalized,and it is proved that a matrix with determinant is det(F)=hq·r(z1)and hq(z1)|dl-r+1(F(z1,z2))is equivalent to its Smith form(Il-r hq((z1)·Ir))if and only if all(l-r)×(l-r)subforms of this matrix have no common zeros on K2Second,study the equivalence problem of a class of n-D(n≥3)polynomial matrix F(z)whose determinant is(z1-f1(z2,…,zn))r(z2-f2(z3,…,zn))r=hr(z)and h(z)|dl-r+1(F(z)).A necessary and sufficient condition for proving the equivalence of this class of polynomial matrix F(z)with its Smith form(Il-r h(z)·Ir)is that all(l-r)×(l-r)subforms of this matrix have no common zeros on Kn.Using recursion,generalized the result to the case where the determinant of the matrix F(z)is det(F)=hq(z)and h(z)|dl-r+1(F(z))is equivalent to its Smith form(Il-r hq/r(z)·Ir)),where q/r is a non-negative integer.The conditions given in this paper can make the operation of the corresponding multi-dimensional system easy.Compared with the previous methods,the tests of these conditions given in this paper are relatively easy to implement,and can be discriminated by calculating the approximate Gr(?)bner basis of the corresponding polynomial ideal,and the corresponding specific equivalent examples are also given to illustrate.