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Fourth order linear differential equations with imprimitive group

Delphine Boucher, Philippe Gaillard, Felix Ulmer
2003 Proceedings of the 2003 international symposium on Symbolic and algebraic computation - ISSAC '03  
In this paper we study fourth order linear differential equations whose differential Galois groups are imprimitive.  ...  This is the lowest possible order where imprimitive non monomial groups occur.  ...  Let L(y) = 0 be an irreducible fourth order linear differential equation whose differential Galois group is an imprimitive non monomial subgroup of SL(4, C).  ... 
doi:10.1145/860854.860871 dblp:conf/issac/BoucherGU03 fatcat:7fwajrsckzg2lpypmycocmng2m

Page 9849 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
differential equations with imprimitive group.  ...  Summary: “In this paper we study fourth order linear differential equations whose differential Galois groups are imprimitive. We derive optimal bounds for the degree of the minimal polynomia!  ... 

Page 1797 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
In the present paper, using invariant theory, the author provides a decision procedure to compute certain unimodular Galois groups of fourth-order homogeneous linear differential equations.  ...  Kovacic’s 1978 algorithm to compute Liouvillian solutions of second-order linear differential equations spurred deep interest in the algorithmic approach to studying linear differential equations.  ... 

On second order homogeneous linear differential equations with Liouvillian solutions

Winfried Fakler
1997 Theoretical Computer Science  
We determine all minimal polynomials for second order homogeneous linear differential equations with algebraic solutions decomposed into invariants and we show how easily one can recover the known conditions  ...  on differential Galois groups [ 10,17,24] using invariant theory.  ...  Let L(y) = 0 be an irreducible second order linear dtfferential equation with an imprimitive unimodular Galois group Y(L).  ... 
doi:10.1016/s0304-3975(97)00055-8 fatcat:ibjxz4wnvrbknnbmwxytylovia

The September meeting of the San Francisco Section

G. A. Miller
1905 Bulletin of the American Mathematical Society  
DICKSON : " The abstract group simply isomorphic with the general linear group in an arbitrary field." (10) Professor L.E.DICKSON : " The abstract group simply isomorphic with the symmetric group." (11  ...  ARTHUR EANUM : " The representation of linear fractional congruence groups with a composite modulus as permutation groups." (6) Professor E. J.  ...  As corollaries are deduced symmetric definitions of the abstract groups isomorphic with the linear fractional and linear non-homogeneous groups on n -1 variables. 10.  ... 
doi:10.1090/s0002-9904-1905-01295-7 fatcat:b372fzsdn5g2hb5vxcb6dtt5zq

The October meeting of the American Mathematical Society

F. N. Cole
1900 Bulletin of the American Mathematical Society  
SMITH : " Geometry within a linear spherical complex.'' (9) Dr. E. J. WILCZYNSKI : " Invariants of systems of linear differential equations."  ...  Wilczynski has shown that the most general point transformation which converts a general system of n linear homogeneous differential equations into another of the same form and order is Equations (1) define  ... 
doi:10.1090/s0002-9904-1900-00769-5 fatcat:xilgrrxdw5akvk4cdlqtmcwabm

Irreducible Linear Differential Equations of Prime Order

Felix Ulmer
1994 Journal of symbolic computation  
With the exception of a nite set of nite di erential Galois groups, if an irreducible linear di erential equation L(y) = 0 of prime order with unimodular di erential Galois group has a Liouvillian solution  ...  , then all algebraic solutions of smallest degree of the associated Riccati equation are solutions of a unique minimal polynomial.  ...  Let L(y) be a linear di erential equation of prime degree p over k with imprimitive di erential Galois group G(L) SL(p; C ).  ... 
doi:10.1006/jsco.1994.1055 fatcat:xyhehgzqh5htfasyadax2e47vu

Galois Groups of Second and Third Order Linear Differential Equations

Michael F. Singer, Felix Ulmer
1993 Journal of symbolic computation  
Using the representation theory of groups, we are able to give simple necessary and su cient conditions regarding the structure of the galois groups of second and third order linear di erential equations  ...  These allow us to give simple necessary and su cient conditions for a second order linear di erential equation to have liouvillian solutions and for a third order linear di erential equation to have liouvillian  ...  Factorization of linear differential equations Let k be a eld and be a derivation on k.  ... 
doi:10.1006/jsco.1993.1032 fatcat:lz4xcsl4p5bh5prfszc5gzague

A symplectic Kovacic's algorithm in dimension 4 [article]

Thierry Combot, Camilo Sanabria
2018 arXiv   pre-print
Let L be a 4th order differential operator with coefficients in K(z), with K a computable algebraically closed field.  ...  Moreover, using Klein's Theorem, algebraic solutions are given as pullbacks of standard hypergeometric equations.  ...  The solutions to a linear ordinary differential equation are all Liouvillian if and only if the connected component of its differential Galois group is solvable.  ... 
arXiv:1802.01023v1 fatcat:thp4fhn2qzhldjnxr72kf4heui

An Algorithm Determining the Difference Galois Group of Second Order Linear Difference Equations

P.A. Hendricks
1998 Journal of symbolic computation  
This algorithm can be considered as an analogue for second order linear difference equations of Kovacic's algorithm for second order linear differential equations (see Kovacic (1986) ).  ...  The solutions of a second order difference equation with an imprimitive difference Galois group are not considered to be 'closed form ' solutions in Petkovsek (1992) .  ...  Hence, the difference Galois group is the imprimitive group where c n = n 2 −1 k=1 2k if n is even and c n = 0 if n is odd and d n = n−1 2 k=1 (2k − 1) if n is odd and d n = 0 if n is even.  ... 
doi:10.1006/jsco.1998.0223 fatcat:hwazgv55pvakxmt5dlw6g2f7w4

Liouvillian Solutions of Linear Differential Equations of Order Three and Higher

Mark Van Hoeij, Jean-François Ragot, Felix Ulmer, Jacques-Arthur Weil
1999 Journal of symbolic computation  
Singer and Ulmer (1997) gave an algorithm to compute Liouvillian ("closed-form") solutions of homogeneous linear differential equations.  ...  We specifically 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)).  ...  This research was partially supported by the European Cathode II working group and, for the third author, also by the Calife project (INTAS-RBFR 95-0412).  ... 
doi:10.1006/jsco.1999.0316 fatcat:h6h5d6lnlvgprh4737dgkolanm

An algorithm for computing a standard form for second-order linear q-difference equations

Peter A. Hendriks
1997 Journal of Pure and Applied Algebra  
In this article an algorithm is presented for computing a standard form for second order linear q-difference equations.  ...  This standard form is useful for determining the q-difference Galois group and the set of Liouvillian solutions of a given equation. @ 1997 Elsevier Science B.V.  ...  For analogous algorithms for second-order differential equations and second-order difference equations we refer to [3] and [l] , respectively.  ... 
doi:10.1016/s0022-4049(97)00017-0 fatcat:sq2vtts54ndxrci2p3eahbok6m

The April meeting of the San Francisco Section

H. C. Moreno
1911 Bulletin of the American Mathematical Society  
KUSCHKE : " The equations of the 10th degree, irreducible in a given rational domain, whose groups are transitive with two as well as five systems of imprimitivity " (preliminary communication). (7) Professor  ...  The most general transformation is found which converts a linear partial differential equation in three variables, with variable coefficients, into another equation of the same form. 9.  ... 
doi:10.1090/s0002-9904-1911-02097-3 fatcat:oq27cs32avdmbczflrxblfvgiu

Note on Kovacic's Algorithm

FELIX ULMER, JACQUES-ARTHUR WEIL
1996 Journal of symbolic computation  
Algorithms exist to find Liouvillian solutions of second order homogeneous linear differential equations (Kovacic, 1986, Singer and Ulmer, 1993b) .  ...  rational solutions of some associated linear differential equations.  ...  The third order case is treated in Singer and Ulmer (1993b) . Definition 1.2. Let L(y) = 0 be a homogeneous linear differential equation of order n with coefficients in k.  ... 
doi:10.1006/jsco.1996.0047 fatcat:owfh7g2a7jaaxawcy3jwwgwef4

Third report on recent progress in the theory of groups of finite order

G. A. Miller
1907 Bulletin of the American Mathematical Society  
and such that to each equation of the group 0.0.= O h corresponds an equation e { e. = e k of the algebra."  ...  KLEIN, Göttingen : " Concerning the connection between the so-called theorem of oscillation of differential equations and the fundamental theorem of automorphic functions." *Burnside, Proc.  ... 
doi:10.1090/s0002-9904-1907-01574-4 fatcat:dmpd5kos7nbm5nx7ncoc5wwdg4
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