Special Linear Groups Of Valuation Ring And Polynomial Matrix

Algorithmic algebra and symbolic computation have important scientific significance and wide application in mathematical theory and engineering calculation,equivalence and factorization of multidimensional matrices are important directions for the cross-fusion of algorithm algebra and symbolic computation.This paper focuses on special linear groups of valuation ring and polynomial matrix,the main contents of this article are as follows:Firstly,in the third chapter,according to the Hermite ring conjecture put forward by Lam,we study the open problem that the valuation ring is Hermite ring.We prove that for any valuation ring V,w=（f（x）,g（x）,e（x））∈ V[x]is unimodular row vector,which can be embedded in a reversible square matrix,we prove that the Hermite ring conjecture holds for general valuation ring V,That is,V is a valuation ring,then V[x]is a Hermite ring.Secondly,in the fourth chapter,we discuss the problem of matrix factorization over valuation rings and arithmetical rings,for any valuation ring V,proved SL_n（V[x]）=E_n（V[x]）（n≥3）,namely,the special linear group on V[x]is consistent with the third elementary commutative group on R[x].Then,for any arithmetical ring R,where n ≥ 3,we show SL_n（R[x]）=SL_n（R）·E_n（R[x]）,every matrix σ（x）∈ SL_n（R[x]）is congruent to σ（0）modulo E_n（R[x]）.Third,in the fifth chapter,we discuss the equivalence of multivariate polynomial matrix.According to the Serre’s reduction of any one dimensional polynomial matrix,the hierarchical recursion can be carried out,and then the Serre’s reduction of multidimensional matrix is studied,we use the theory of modules and Grobner of modules to prove a multidimensional matrix A,if A～diag（1,B₁）～diag（1,1,B₂）,then B₁ is not equivalent to diag（1,B3）,and give also an example that a 4-D polynomial matrix is not equivalent to its Smith form.