Reproduction of Fuchs Relation for the Group 2 4 SL S by Using Groebner Bases

In 1872, Lazarus Fuchs used a new tool which is The Invariant Theory to construct the minimal polynomial of an algebraic solution of a differential equation of second order. He expressed the coefficient of the equation in terms of the (semi-)invariants of its differential Galois group. In this paper we will give another method to obtain Fuchs Relation: 2 2 0 2 1 36 6 5 5 I a S SS S w            for the octahedral groupe 2 4 SL S by using Groebner Basis; a tool which is introduced in 1965 nearly two century after


Introduction
This section is a recall of some definitions in the theories that we will depond on in our study.
In the second section we will give an Algorithm to obtain Fuchs Relation and in the third section we will expose the codes Maple and Magma we have used.
In 1872, Lazarus Fuchs used a new tool which is the Invariant Theory to construct the minimal polynomial of an algebric solution of a differential equation of second order whose solutions space is generated by   12 , yy (Baldassarri, F. & Dwork, B., 1979;Kovacic,. J.. 1985).He expressed the coefficient of the equation in terms of the (semi-)invariants , IS and differential semi-invariant , SS   of its differential Galois group .L. Fuchs found this relation: such that w is the wronskian and 0 a is the coefficient in the ordinairy linear differential equation: I and S are the invariant and the semi-invariant: by the matrices A and B such that: (for more knowledge about Representation Theory refer to (Fulton, W. & Harris, J., 1991), (Weyl, H., 1946)).

Invariants Theory
Definition 1.Let V be a finite dimensional K -vector space and G be a linear subgroup of which remains unchanged under the group action, i.e f f g  for all gG  .If, for some gG  , f and fg differ from each other only by a constant factor then the polynomial function f is called a semi-invariant (or a relative invariant) (for more knowledge about Invariant Theory refer to (Benson, D. J., 1993), (Cox, D. J., 1992), (Sturmfels,. B., 1993)).
Remark.The computation of Reynold Operator and Molien series which is needed to find invariants is implemented in different Programs such as Mathematica and Magma.

Differential Galois Theory Definition
Let C be a field algebraically closed and   , k  a differential field be of characteristic 0 .Consider the following ordinary homogeneous linear differential equation: Ly always exists and is unique up to differential isomorphisms.This extension plays the same role for a differential equation as a splitting field for a polynomial equation.The set of all automorphisms of K , which fix   , k  elementwise and commute with the derivation in K , is a group, (for more knowledge about Differential Galois theory refer to (Magid,. A., 1994), (Kolchin, E. R., 1948)).
Indeed, the differential Galois group of   2 is unimodulair (cf.(Kaplansky, I., 1957), p.41), so we will deal with the form   2 (for more knowledge about Liouvillian solutions of second order homogeneous Linear Differential Equations refer to (Fakler, W., 1996)).

Compute L. Fuchs Relation by using Groebner Bases
In this section, we compute the relation of L. Fuchs by using Groebner Bases (for more knowledge about Groebner Basis refer to (Adams, W., & Loustaunau, P., 1994), (Cox, D. J., 1992)).
Which relates (semi-)invariants S , I , differential semi-invariants S  and S  , the wronskian w and the coefficient 0 a .In fact, we will prove in this paper that we can obtain this relation by using Groebner Bases, in particular the ideal ( (Cox, D. J., 1992), p.115).
Before giving The Algorithm, we will recall this proposition from (Cox, D. J., 1992): x y y where any monomial involving one of 1 , , be the remainder of f on division by G .Then: is an expression of f as a polynomial in 1 , , m f f The Algorithm is as follow: 1. Take the ring  (Fuchs Relation, III).
In this section we provide two codes in Magma and in Maple.
solutions.By extending the derivation  to a system of fundamental solutions and by adjunction of these solutions and their derivatives to   , k  in a way the field of constants does not change, one gets 1 so-called Picard-Vessiot extension (PVE) of   0Ly .With these assumptions, the PVE of   0 linear differential equation over a differential field   , k  whose field of constants C is algebraically closed of characteristic 0, with differential Galois group

Groebner Basis Definition 1. A monomial ordering
Groebner basis of I if and only if the leading term of any element of I is divisible by one of the Groebner basis.Further more, any Groebner basis for an ideal I is a basis of I .G be a Groebner basis of I with respect to lex order where 12 , E. R., 1948) A differential equation   0 Ly with coefficients in k has Only solutions which are algebraic over k if and only if   i .Corollary ((Cox, D. J., 1992), p.77).Fix a monomial order.Then every ideal let 1