Research On Grobner Bases And Primary Decom-position Of Ideals And Matrix Factorization

Primary decomposition of ideals and matrix factorization are two key problems of computing Algebra. They have wide range of applications in the study of computer Algebra, computing Algebraic geometry, Algebraic coding and cryptography, multidimensional systems and so on. The theory of Grobner bases is an important tool in the studying of primary decomposition of ideals and matrix factorization.The thesis is divided into six chapters. We mainly study primary decomposition of ideals and matrix factorization and the application of Grobner bases.The first chapter is devoted to surveying research background, results, methods in this area.In the second chapter, we mainly discuss and study algorithm for computing Grobner bases of ideals in polynomial ring over principal ideal domain (PID). We extend the GVW algorithm to polynomial ring over PID, which is the most efficient and simple algorithm computing Grobner bases of ideals in polynomial ring over field. We also give example to explain the extended algorithm.In the third chapter, firstly, we study the problem of general multivariate (n-D) polynomial matrix factorizations. For F∈Cl×m[z], we concern theι-1order minors of F in conditions, obtain some classes of n-D polynomial matrices admit the general matrix factorization. Then, we discuss factor prime factorization for n-D polynomial matrixes, and obtain some further simplifications conditions to check a divisor regular with respect to F. Finally, we study matrix prime factorization over PID. We obtain that a full row rank F∈Dl×m [z] has a minor prime factorization if and only if p(F):d is a free module of rankι whereρ(F)is the submodule generated by the rows of F. When f is a regular with respect to F, we also prove that F has a factor prime factorization if and only if ρ(F):d is a free module of rankι.In the fourth chapter, firstly, we study whether a given zero-dimensional ideal is in normal position with respect to a variable or not, give an equivalence condition of the ideal is in normal position with respect to a variable or not. We obtain a reduced method for primary decomposition of the ideal. For some kinds of ideals, parts of their primary components can be obtained quickly. Furthermore, when the given zero-dimensional ideal I is not normal with respect to every variable, we discuss the selection of c for the extension ideal J of I, and find a specifically and quick method of selecting c, which is remove off the randomization.In the fifth chapter, we mainly study some kinds of Grobner bases with special properties. The definition of weak Grobner basis is given, and its applications to polynomial composition are presented in this chapter. We also use the new criterion which we obtain to study the following two problems.Problem1When do all the subsets of the form {f+s,g+t} form Grobner bases? Where s and t are arbitrary elements in the coefficient field.Problem2When do all the subsets of the form{fλ, gσ} form Grobner bases? Where λ and a are arbitrary non-negative integers.We presented that all the subsets{f+s,g+t} are Grobner bases if and only if ιm(f)and ιm(g)are relatively prime, and all the subsets {fλ,gσ} are Grobner bases if and only if ιm(f) and ιm(g) are balanced, where λ and σ are arbitrary non-negative integers.In the sixth chapter, we obtain the necessary and sufficient condition for f1,f2,…,fm∈R[x1,…,xn] are mutually prime by using the theory of resultants.