Residual ideals of MacLane valuations

Julio Fernández, Jordi Guàrdia, Jesús Montes, Enric Nart

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. Let (K,v) be a discrete valued field and let x be an indeterminate. In 1936, MacLane determined all valuations on K(x) extending v. His work has been reviewed and generalized by Vaquié, by using the graded algebra of a valuation. We extend Vaquié's approach by studying residual ideals of the graded algebra as an abstract counterpart of certain residual polynomials having a key role in the computational applications of the theory. This enables us to determine the structure of the graded algebra of the discrete valuations on K(x). Also, let P be the set of monic irreducible polynomials with coefficients in the completion Ov of the valuation ring of v. The constructive methods of the paper yield a canonical bijection P/≈→M, between the set of Okutsu equivalence classes of prime polynomials and a certain MacLane space M, whose points may be described in terms of discrete parameters associated with valuations on K(x).
Original languageEnglish
Pages (from-to)30-75
JournalJournal of Algebra
Volume427
DOIs
Publication statusPublished - 1 Apr 2015

Keywords

  • Key polynomial
  • MacLane chain
  • Newton polygon
  • Okutsu invariants
  • Primary
  • Residual ideal
  • Residual polynomial
  • Secondary
  • Valuation

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