Journal Article10.4153/CJM-1995-041-9
A Simple Algorithm for Deciding Primes in K[[x,y]]
TL;DR: In this paper, a generalised version of Hensel's Lemma is developed for the proofs of the Tschirnhausen transformation, where K is an algebraically closed field of characteristic 0.
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Abstract: The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.
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Citations
Valuations in algebraic field extensions
TL;DR: In this paper, the authors give an explicit description of the limit key polynomials, which can be viewed as a generalization of the Artin-Schreier polynomial.
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Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields
Adrien Poteaux,Marc Rybowicz +1 more
- 24 Jun 2015
TL;DR: This paper reduces this bound to O~(d4+d2log q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps.
Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
TL;DR: In this paper, the Pierce-Birkhoff conjecture was proved for regular 2-dimensional rings with dimension 2 by using approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of Q).
Resolution of Singularities: An Introduction
Mark Spivakovsky,Mark Spivakovsky +1 more
- 01 Jan 2020
TL;DR: A survey of singularity resolution can be found in this article, where the main topics covered are the early days of resolution (fields of characteristic zero and dimension up to three), Zariski's approach via valuations, Hironaka's celebrated result in positive characteristic (mostly up to dimension three), de Jongs approach via semi-stable reduction, Nash and higher Nash blowing up, as well as reduction of singularities of vector fields and foliations.
Irreducibility criterion for algebroid curves
TL;DR: A new notion of local tropical variety is introduced which is a straightforward extension of tropism introduced by Maurer, and the irreducibility criterion for algebroid curves is given in terms local tropical varieties.
References
Generalized Newton-Puiseux Theory and Hensel's Lemma in C[[x, y]]
TL;DR: This paper introduces a method which is parallel to the classical Newton-Puiseux theory, yet avoids blowing-ups and fractional power series, except in the proofs.