Compound. The Noether Whole Ring Groebner Bases

Composition is the operation of replacing variables in a polynomial with other polynomials. First, we sum up the history and the study background of Groebner basis, composed Groebner basis theory and introduce the basic knowledges of Groebner basis theory and the relative Algebra theory. Then, we present the new studies of scholars on composition and Groebner basis properties, and detaily introduce the commutativity between Groebner basis and composition, homogeneous composition, weighted homogeneous composition and the reduced composed Groebner basis over field.By using S-polynomial and syzygy condition, this paper presents and proves a sufficient condition for Groebner bases computation in n variables in a polynomial ring over Noetherian domain and composition in m (m (?) n) variables in another polynomial ring over the same Noetherian domain is commutative: composition is compatible with two term orderings on the two different polynomial rings and composition is a list of monic polynomials with its leading powering products is the products of permuted powering and powering products of other remained variables. Therefore, minimal Groebner bases computation is also commutative with composition. Especially, Groebner bases computation and composition is commutative if composition is compatible with term orderings and composition is a list of monic polynomials with its leading powering product is a permuted powering. Then, for term orderings of different properties, we present and prove a sufficient condition for Groebner bases computation in a polynomial over Noetherian domain under some term ordering is commutative with composition under another term ordering: composition is a list of monic polynomials with its leading powering product is a permuted powering.