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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


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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
1999 Academic Press
In this paperknerelaiteton®idawidellnotsnaste¯eldofc¯eldwhosCis algebraically closed of characteristic 0. For the computations, however (Section 2.2), we will consider moreconcretelythe¯eldC(x) with the usual derivationddx. We denote byL=n+ an¡1+¢ ¢ ¢+a00(aik,Lk[]db)yadoraneenrtaitai®leorpL(y) =y(n)+ n¡1 an¡1y(n¡1)+¢ ¢ ¢+a0ythe corresponding linear homogeneous di®erential equation.= 0 Wenowbrie°yrecallthebasicde¯nitionsfromdi®erentialGaloistheorythatareneeded later on (see Magid, 1994; Singer, 1997; van der Put, 1998, for details and references). APicard–Vessiot extension(PVE)KofkforLoinetsnldexal¯eentii®erdasiK=k < y1     yn>, where{y1     yn}a fundamental set of solutions, with no new constants inKe¯dlitgnofrntlevauiitplasofI.qeehtsitL(y) = 0. Under our assumptions a PVEexistsandisuniqueuptodi®erentialautomorphisms.WedenotebyV(L) the solution spaceV(L) ={yK|L(y) = 0}of an operatorL. The dimension ofV(L) equals the order ofL. Thep®iredGaaltienougrisloGofLtsiopuorgehitladf®irene
E-mail:hoeij@math.fsu.edu kE-mail:ragot@unilim.fr, weil@unilim.fr ¤E-mail:ulmer@univ-rennes1.fr
0747–7171/99/000001 + 21
$30.00/0
c 1999 Academic Press
2
M. van Hoeijet al.
automorphisms ofKk. It acts faithfully on the vector spaceV(L), and soGcan be viewed as a subgroup ofGL(V(L)); more precisely, it is a linear algebraic group over C. There is a Galois correspondence between algebraic subgroups ofG®iddnareneitla sub¯elds of the PVE ofL(y) = 0. The ¯xed ¯eld ofGunder this correspondence isk. A solution ofL(y) = 0 inkis called arational solution, a solution in an algebraic extension ofkis called analgebraic solution, a solution whose logarithmic derivative is inkis called anexponential solutiontulobnoinolegnigadtoeri®tien¯ealdldnsaa obtained by successive adjunctions of integrals, exponentials of integrals and algebraic extensions is called aLiouvillian solution. IfLhas a Liouvillian solution thenLalso has a Liouvillian solution of the formy= exp(Rr) whererkis algebraic overk(see Singer, 1981, 1997).
Definition 1.1.An elementrof the PVE is called aRiccati solutionforLifris the logarithmic derivativer=y0yof some non-zero solutionyofL. Ifris an algebraic Riccati solution (a Riccati solution ink) then the minimum polynomial ofroverkis called aRiccati polynomialofL.
Definition 1.2.An elementISymm(V(L)) is called asemi-invariantof the di®er-ential Galois groupGofLof degreemif there is a characterÂ:G7→ Cof degree 1 such that for allgGthe action ofgonIisg(I) =Â(g)I. IfÂis the trivial character, i.e. gG g(I) =I, thenIis called aninvariant.
We call a polynomial or semi-invariantcompletely factorableif it is a product of linear factors. It is known (see Section 2 in Singer and Ulmer, 1997) that an algebraic Riccati solutionrof degreemoverkcorresponds to a completely factorable semi-invariantI of degreemplaaGneitrguoolsioferi®edthG. In Singer and Ulmer (1997), the Riccati polynomial ofris computed as follows:
(1) Compute the spaces of all semi-invariants up to a certain degree. This degree can be bounded in terms of the order of the operator using group theory. (2)Ineachsuchspace,¯ndacompletelyfactorablesemi-invariantISymm(V(L)) wheremis the degree. If such a semi-invariant is found, thenL(y) = 0 has an alge-braic Riccati solution and the ¯rst coe±cient of the associated Riccati polynomial is given by the logarithmic derivative of the value ofI. The value ofIis the image ofIinK. (3) In Singer and Ulmer (1997), the factorization ofIinto linear forms is used to computetheremainingcoe±cients.Thisinvolvescomputingwithsplitting¯elds and so this step could be costly.
The main objective in this paper is to give an e±cient method to compute (if it exists) a Riccati polynomial. The goal is a method that is e±cient enough to handle operators of order three on a computer, or sometimes even higher order if the Riccati polynomial is not too big. First we extend the algorithm in van Hoeij and Weil (1997) for computing invariants to the case of semi-invariants. If a completely factorable semi-invariant is known and given inacertainform,thenTheorem2.1immediatelygivesusallcoe±cientsoftheRiccati polynomial. This central result avoids the third step above and makes the computation much more feasible in practice.