Arithmetical Rank Of Ideals

The arithmetical rank is a kind of important tools in the study of commutative algebra and algebraic geometry, I mainly prove three parts in this paper.Firstly, On the basis of the method provided by Schmitt and Vogel in the literature [1],which discusses the calculation of upper bound of arithrmetical rank on certain condition,Barile issued a method of calculation of upper bound of arithrmetical rank of monomial ideals in algebraic closed field in the literature [2]. In this paper I discuss the calculation method of upper bound of arithrmetical rank of polynomial ideals in algebraic closed field.Secondly, in the literature [3], Barile provides the size of arithrmetical rank of edge ideal of the graph which contains two circles which have only one common intersection point, and proves the arithretical rank of edge ideal is equal to the projective dimension of R/I(G). And in this paper I illustrate the upper bound of the arithrmetical rank of edge ideal of the graph contains three circles which have two common intersection points,furthermore I illustrate the existence of this upper bound and the result of ara I(G) =pd R/I(G) by examples.Thirdly, in the literature [5], Olteanu provides the conclusion that the arithrmetical rank of lexsegment ideal is equal to projective dimension when the degree of monomial generator is two. And in this paper, I discuss several conclusions on the condition that the arithrmetical rank of some lexsegment ideals equals the projective dimension of R/I(G)when the degree of monomial generator is three.