Counting the number of distinct distances of elements in valued field\n extensions

2017/05/26 by Anna Blaszczok, Franz‐Viktor Kuhlmann, Blaszczok, Anna +1
Computer Science · Mathematics · #12J10 #12J25 #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1705.09541

openalex publication_date 2017/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The defect of valued field extensions is a major obstacle in open problems in\nresolution of singularities and in the model theory of valued fields, whenever\npositive characteristic is involved. We continue the detailed study of defect\nextensions through the tool of distances, which measure how well an element in\nan immediate extension can be approximated by elements from the base field. We\nshow that in several situations the number of essentially distinct distances in\nfixed extensions, or even just over a fixed base field, is finite, and we\ncompute upper bounds. We apply this to the special case of valued functions\nfields over perfect base fields. This provides important information used in\nforthcoming research on relative resolution problems.\n

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