Some Properties Of The Coordinate Ring Over Finite Fields And Its Triangular Compositions

The relationship between polynomial equation set and its corresponding affine variety over finite fields plays a fundamental role in many important fields. In this paper, we develop the ’Hilbert s Nullstellensatz theorem which is the important result. The research of the coordinate rings over finite fields has very important significance such as in coding theory, the corresponding triangular decomposition in coordinate rings and Nullstellensatz theorem also has crucial significance in theoretical research and practical applications.In the theory of coordinate ring over finite fields, we can easily find that any nonempty subset inn qF is an affine variety. In this paper, our starting point is the special nature over finite fields, and based on the study of polynomial function, we discuss contents as follows:Firstly, we discuss the property of the coordinate ring over finite fields, give the Hilbert’s Nullstellensatz theorem over finite fields. The new theorem no longer required to be in algebra closed fields. Furthermore, we present some applications of the new theorem.Secondly, based on the proof that the affine varieties of any ideal in the polynomial rings can be represented by the affine varieties generated by a single polynomial, we prove that the affine varieties of any ideal in the coordinate rings can be represented by the affine varieties generated by a single polynomial function. The polynomials and polynomial functions that satisfy the above demands are constructed by the properties of the irreducible polynomials and the properties of its homogenization. In addition, based on the special properties of the finite fields, we construct polynomial functions satisfying the single representation of the affine varieties in coordinate rings using another method. At the same time, we give the corresponding algorithms and some examples.Finally we promote the basic knowledge of the triangular decomposition from ploymomial rings to coordinate rings over finite fields, and we give an upper bound of the length of the strictly lower triangle set. Triangular decomposition algorithm in polynomial function set is given on the base of triangular decomposition algorithm in polynomial set. Some example are given to illustrate our algorithm.