2023/08/10 by Razieh Ahmadian, Ahmadian, Razieh, Steven Dale Cutkosky +1
Mathematics · #12J20 #14B25 #16W60 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2308.05656
openalex publication_date 2023/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose that (K,v0) is a valued field, f(x)∈ K[x] is a monic and irreducible polynomial and (L,v) is an extension of valued fields, where L=K[x]/(f(x)). Let A be a local domain with quotient field K dominated by the valuation ring of v0 and such that f(x) is in A[x]. The study of these extensions is a classical subject. This paper is devoted to the problem of describing the structure of the associated graded ring \rm grv A[x]/(f(x)) of A[x]/(f(x)) for the filtration defined by v as an extension of the associated graded ring of A for the filtration defined by v0. We give a complete simple description of this algebra when there is unique extension of v0 to L and the residue characteristic of A does not divide the degree of f. To do this, we show that the sequence of key polynomials constructed by MacLane's algorithm can be taken to lie inside A[x]. This result was proven using a different method in the more restrictive case that the residue fields of A and of the valuation ring of v are equal and algebraically closed in a recent paper by Cutkosky, Mourtada and Teissier.