| In this paper we mainly study problems about ideal theory,which include primary decomposition of ideals,properties and computation for polynomial composition and Gr(o|¨)bner bases.Let k[x1,…,xn]be a polynomial ring in the variables x1,…,xn with coefficient from field k.I is a zero-dimensional ideal in k[x1,…,xn].The primary decomposition of I is decompose I to the intersection of some primary ideals.(?)=(θ1,…,θn) be a list of polynomials in k[x1,…,xn].Composition of f(x1,…,xn) by(?) is usedθ1,…,θn replace variables x1,…,xn obtain f(θ1,…,θn),denoted by fo(?).Firstly,we discuss and study primary decomposition problem of zero-dimensional ideals in Neother ring.After study properties of zero-dimensional ideals and ideals in Neother ring,we discuss the relation between normal position and general position of I,and prove that they are equivalent on some condition.We can find a list of c= (a1,…,an-1) that make the extension ideal J of I is normal position with respect to a variable z andφc(I) is general position with respect to the lexicographic order x1＞x2＞…＞xn when I is not normal with respect to every variable.Discussing the selection of c,we find a more efficient method than[2].Then,we find a method to judge a zero-dimensional ideal is prime or primary,which is use Gr(o|¨)bner basic directly,and need not the expensive condition of rood operation or normal position about a variable.Secondly,we study the problem of primary decomposition of zero-dimensional ideals over finite field.On the basis of professor Gao's research on this problem,we give a general algorithm to primary decomposition of this kind of ideal.Then we discuss the method which Gao proposed,and when it is separable that the basic element of the invariant subspace of the ideal's quotient ring,we give necessary and sufficient condition to judge it.Then we obtain an improved algorithm to primary decomposition of this kind of ideal.We also get an approach to judge whether this kind of ideal is primary ideal or not,which only need compute the rank of the matrix.Finally,we study properties and computation for polynomial composition with H-Gr(o|¨)bner bases and W-Gr(o|¨)bner bases.We obtain equivalent condition of computation commuting of them respectively.We also discuss application of this two Gr(o|¨)bner bases.
|
PDF Full Text Request