Zero-dimensional Algebraic Varieties Ideal Gr (?) Bner Bases Of And Characteristics Of The Columns And Their Applications

Zero-dimensional algebraic varieties are one of the main topics in computational algebraic geometry research field. There are many applications in modern geometry, algebra and engineering.The article introduces the background of zero-dimensional algebraic varieties and presents status of international research. Then it explains the basic concepts and related theories, such as algebraic affine varieties, Gr(o|¨)bner bases of ideals and triangular series.Gr(o|¨)bner bases and triangular series are the main tools in the research of algebric varieties. This paper introduces a new approach to compute Gr(o|¨)bner bases by vanishing ideals of finite sets of points by determinant, which has good forms and conciseness, so that it can more efficient to calculate the greatest common factor of polynomials in any field.The construction of polynomial interpolation is also introduced by determinant and Gr(o|¨)bner bases.Based on zero-dimensional algebraic varieties, the main results of this paper are:1. The present method to construct Gr(o|¨)bner bases of zero-dimensional algebraic varieties is improved by applying algebraic affine varieties and the theory of monomial bases in order to have better forms and more concise.2. Let K be a field and P1 ,L ,Pn be distinct points in K m, the Gr(o|¨)bner bases of zero-dimensional algebraic varieties ideals are constructed by determinant applying the relationship between algebraic affine varieties, polynomials and ideals. then we can not only calculate this monomial bases but also constructe polynomial interpolations in multiple variables.3. A new approach to calculate the greatest common factor of univariate polynomial by Gr(o|¨)bner bases.4. Several examples are given to illustrate the new method which has better forms and fast calculation.