Linear differential operators for polynomial equations
type: Item
instance of: scholarly article
title: Linear differential operators for polynomial equations
zbMATH Open document ID: 1030.12002
author: Olivier Cormier
author: Barry Trager
author: Felix Ulmer
author: Michael F. Singer
published in: Journal of Symbolic Computation
publication date: 2003-03-23T00:00:00Z
review text: Let \(k_0\) be a number field and \(\overline{k_0}\) be its algebraic closure. Let \(P\in k_0(x)[y]\) be a squarefree polynomial in \(y\). The derivation \(\delta=\frac d{dx}\) extends uniquely to the algebraic closure \(\overline{k_0(x)}\) of \(k_0(x)\). The minimal operator associated with \(P\) is defined as the monic differential operator \(L_P=\delta^t+a_{t-1}\delta^{t-1}+\dots+a_0\) with \(a_i\in k_0(x)\) of smallest positive order such that \(L_P(y)=0\) for all roots of \(P\) in \(\overline{k_0(x)}\). Some algorithms for calculating this operator are given. This allows one to calculate the genus of the complex curve defined by \(P=0\) (when \(P\) is absolutely irreducible), the absolute factorization of \(P\) over \(\overline{k_0}\), and calculate information concerning the Galois group of \(P\) over \(\overline{k_0}(x)\) as well as over \(k_0(x)\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 1886757
zbMATH Keywords: complex curve
zbMATH Keywords: genus
zbMATH Keywords: differential operator
zbMATH Keywords: absolute factorization
zbMATH Keywords: Galois group
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
full work available at URL: jsco.2002.0564
OpenAlex ID: W1998088789
cites work: Q3471654
cites work: Factoring Rational Polynomials over the Complex Numbers
cites work: On rational solutions of systems of linear differential equations
cites work: On solutions of linear ordinary differential equations in their coefficient field
cites work: On expansion of algebraic functions in power and Puiseux series. I
cites work: On expansion of algebraic functions in power and Puiseux series. II
cites work: Calcul pratique des coefficients de Taylor d'une fonction algébrique
cites work: On Transitive Permutation Groups
cites work: Q4827952
cites work: Computing the Galois group of a polynomial using linear differential equations
cites work: Absolute Factorization of Polynomials: A Geometric Approach
cites work: Q5531597
cites work: Q5577154
cites work: Complexity of computation of embedded resolution of algebraic curves
cites work: Endliche Gruppen I
cites work: Effective Noether irreducibility forms and applications
cites work: Q5287502
cites work: Algorithms in Algebraic Number Theory
cites work: Computation of Galois groups over function fields
cites work: Q4263461
cites work: Special issue: Algorithmic methods in Galois theory
cites work: Polynomial and rational solutions of holonomic systems
cites work: Q3264980
cites work: Associative differential operations
cites work: Gröbner deformations of hypergeometric differential equations
cites work: Testing reducibility of linear differential operators: A group theoretic perspective
cites work: Galois groups of second and third order linear differential equations
cites work: The Computational Complexity of the Resolution of Plane Curve Singularities
cites work: Q5789039
cites work: An algorithm for computing an integral basis in an algebraic function field
cites work: Factorization of differential operators with rational functions coefficients
cites work: Modern computer algebra
cites work: Q3287734
cites work: Polynomial algorithms in computer algebra
cites work: Q4274348
DOI: 10.1006/JSCO.2002.0564
description: scientific article; zbMATH DE number 1886757