Invariant Theory And Application

The group action plays a significant role in algebra. Some research in algebra canusually begin with the group action. Particularly, the invariant theory derived from theaction of group on the polynomial ring becomes an more important field. Moreover, therelated knowledge of invariant been applicated to kinds of areas, such as computervision, material science, geometric classification.Considering the polynomial system as the main study object, the following termsare mainly discussed in this paper:1. The knowledge of the zero dimensional ideals’ primary decomposition is appliedin this paper. The irreducibility of polynomials are decided in the following: Firstly, bycomputing the rank of the related square matrix under the linear transformation, adetermine condition about the polynomial reducible of the single variable over finitefields is presented. Secondly, by introducing the related properties of resultant, asufficient and necessary condition about the reducibility over finite fields is offered.Besides, though the primitive polynomial is also irreducible, an improved method aboutthe determination condition of the primitive polynomial is presumed.2. On the one hand, some problems on the division of a finite set are discussed byapplying the group action. On the other hand, the invariant membership problem ispresented from the two aspects: the finite generation of group and the homogeneousrepresent of a polynomial.3. In the ninetieths, the Derksen ideal presented by Derksen, which plays animportant role in the invariant theory. Moreover, some results about homomorphism onpolynomial ring are introduced, so that the Derksen idea can be dealt withhomomorphism of polynomial. And some new results in this topic are proposed.