2015/02/13 by Ye Liang, Liang, Ye
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #cs.SC #math.AC
paper · pdf · doi:10.48550/arxiv.1502.03967
arxiv created 2015/02/13 · openalex publication_date 2015/02/13 · arxiv updated 2015/02/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
When studying local properties of a polynomial ideal, one usually needs a theoretic technique called localization. For most cases, in spite of its importance, the computation in a localized ring cannot be algorithmically preformed. On the other hand, the standard basis method is very effective for the computation in a special kind of localized rings, but for a general semigroup order the geometry of the localization of a positive-dimensional ideal is difficult to interpret. In this paper, we introduce a new ideal operation called extraction. For an ideal I in a polynomial ring K[x1,…,xn] over a field K, we use another ideal J to control the primary components of I and the result β(I,J) is called the extraction of I by J. It is still a polynomial ideal and has a concrete geometric meaning in Kn, i.e., we keep the branches of V(I) ⊂ Kn that intersect with V(J) ⊂ Kn and delete others, where K is the algebraic closure of K. This is what we mean by visible. On the other hand, we can use the standard basis method to compute a localized ideal corresponding to β(I,J) without a complete primary decomposition, and can do further computation in the localized ring such as determining the membership problem of β(I,J). Moreover, we prove that extractions are as powerful as localizations in the sense that for any multiplicatively closed subset S of K[x1,…,xn] and any polynomial ideal I, there always exists a polynomial ideal J such that β(I,J)=(S-1I)c.