Research Of Properties And Computations For Composition And Gr(?)bner Bases

This paper studies properties and computation for composition and Grobner bases.Let K[xi,X2,...,xn] be a polynomial ring in the variables xi,x2,...,xn with coefficient from field K, 0 = {9i,92,.. .,9n) be a list of n polynimials in K[xi,x2,...,xn]. Composition of f{xux2,... ,xn) by 0 is f(9i,92,...,6n), denotes by / o 6. Let F be a non-zero set of polynomial in K[x\,x2,..., xn], we define: FoQ = {foQ\feF}. We say a subset G a Grobner if G is Groebner basis of Ideal(G) generated by G, say a Grobner basis computation commuting with composition by 0 if Go0 is Grobner basis for any Grobner basis, say a term ordering being compatible with composition if VpV q =? polt(0) > <7olt(0).Firstly, we give a new proof of Hong's classical theorem, the proof is simple, and new method. After that, we discuss properties and computations for universal Grobner bases % monomial Grobner bases ? homogeneous Grobner bases ? F-homogeneous Grobner bases under composition. We get equivalent conditions of monomial Grobner basis , homogeneous Grobner basis , F-homogeneous Grobner basis computations commuting with composition by 0. Our results overcome formerly single studing composition and Grobner basis. One enrich content studing composition and Grobner basis. Easpecially, we take the lead in studing some Grobner bases under composition, this is a bigger breach.Secondly, we study compatibilities of composition and term ordering , homogeneous compatibility of composition and term ordering , F-homogeneous compatibility, and give concrete algorithm (judging method). In this way, we resolved Hong's open problem(1998), and made abstrct algebric condition more clear, it is favourable to judgment and application. After that, we study operation of composition, and give concrete forms of composition for some commonterm orderings.Finally, we study Grobner bases and membership problems of ideal for weyl-algebra , Clifford and Grassmann algebra, and get some interesting results. Easpecially, we get clearly Grobner bases of some left ideal in weyl-algebra, provide new method for studing some differential equation systems, we depict the memerbship problem of two-side ideal for binomial polynomial rings.