21 Pages
English

# Article No jsco Available online at http: www idealibrary com on

Description

Niveau: Supérieur, Doctorat, Bac+8
Article No. jsco.1999.2199 Available online at on J. Symbolic Computation (1999) 00, 1{21 Liouvillian Solutions of Linear Difierential Equations of Order Three and Higher MARK VAN HOEIJy{, JEAN-FRANO» IS WEILzk, FELIX ULMERx⁄⁄ AND JACQUES-ARTHUR WEILz yFlorida State University, U.S.A. zUniversit¶e de Limoges, France xUniversit¶e de Rennes I, France Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (\closed-form) solu- tions of homogeneous linear difierential equations. However, there were several e–ciency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semi- invariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semi-invariant that factors into linear forms, one gets all coe–cients of the min- imal polynomial of an algebraic solution of the Riccati equation, instead of only one coe–cient. These coe–cients come \for free as a byproduct of our algorithm for com- puting semi-invariants. We speciflcally detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997)).

• linear difierential

• immediately gives

• kovacic-like algorithm

• invariants then

• algorithm semi

• homogeneous linear

• all semi-invariants up

Subjects

##### State university system

Informations

J. Symbolic Computation(1999)00, 1–21 Article No. jsco.1999.2199 Available online at http://www.idealibrary.com on
LiouvillianSolutionsofLinearDi®erentialEquations of Order Three and Higher
MARK VAN HOEIJ†¶ »OIS, JEAN-FRAN WEIL‡k, FELIX ULMER§¤ JACQUES-ARTHUR WEILFlorida State University, U.S.A. nUrevitisLedeimoges,France §ncra,FsIevireistdeReneenUn
AND
Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form”) solu-tionsofhomogeneouslineardi®erentialequations.However,therewereseverale±ciency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semi-invariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semi-invariant that factors into linear forms, one getsalleic±ostneocheftn-mi imal polynomial of an algebraic solution of the Riccati equation, instead of only one coe±cient. These coe±cients come “for free” as a byproduct of our algorithm for com-putingsemi-invariants.Wespeci¯callydetailthealgorithminthecasesofequationsof order three (order two equations are handled by the algorithm of Kovacic (1986), see also Ulmer and Weil (1996) or Fakler (1997)). In the appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm. c
1. Introduction