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#
# L = mult( Dx + r1 + r2, Dx - r2, Dx - r1, dom ),
#
# where dom = [Dx,x]. Then there exist polynomials a, b
# such that
#
# L = Dx^3 + a*Dx + b.

136
#
# This procedure computes the group of
#
# Lred = LCLM( Dx + r1 + r2, Dx - r2, Dx - r1, dom).
#
# Moreover, if this group is a one-dimensional torus, then
# the procedure computes the matrix representation
# of this group on the solution space of Lred, relative to
# the ordered basis {R1, R2, 1/R1/R2}. Here, R_i is a
# function whose logarithmic derivative is r_i for i = 1,2.
# This ordered basis is chosen because it corresponds
# with a basis of the solution space of L, relative to which
# the group of L is upper-triangular. In particular, R1
# is a solution of L.
#
#
# This procedure returns as output a list of one of
# the following types:
#
#
# [0] -- if the group of Lred is trivial (0-dimensional).
#
#
# [2] -- if the group of Lred is C* x C* (2-dimensional).
# In this case, the group is represented as
#
# { diag( u, v, w ) : uvw = 1}.
#
#
# [1, [e1, e2]] -- if the group of Lred is C* (1-dimensional).
# In this case, the matrix representation
# referred to above has image
#
# { diag(u,v,w): u^e2 = v^e1, uvw = 1 }.
#
# Moreover, it is parameterized by the mapping
#
# t |--> diag( t^e1, t^e2, t^(-e1-e2) )
#
# for t in C*. Also, e1 >= 0; this is
# guaranteed in the procedure by replacing
# (e1, e2) with (-e1, -e2) if necessary.
#
#
local jlist, elist;
if ( rfldtest( r1, x ) and rfldtest( r2, x ) ) then
#
# group is trivial
#

137
return [0] ;
fi;
jlist := ratfun_relation( r1, r2, x );
if ( jlist = [0, 0] ) then
#
# group is C* x C*
#
return [2] ;
fi;
#
# ratfun_relation returns [j1, j2] such that j1*r1 + j2*r2 = hâ€™/h
# for some rational function h, i.e., R1^j1 * R2^j2 = h.
#
# Define [e1,e2] = [-j2, j1], so that replacing R1 with t^e1 * R1
# and R2 with t^e2 * R2 preserves the relation R1^j1 * R2^j2 = h.
# Then make sure that e1 >= 0.
#
elist := [ (-1) * jlist[2], jlist[1] ];
if elist[1] < 0 then
elist := [ (-1)*elist[1], (-1)*elist[2] ];
fi;
[ 1, elist ];
end:

#
# III. o3np code
#

sl2test := proc( L, dom )
#
# Input: L, an irreducible
# second-order differential operator
# over dom whose group is known to be
# either SL_2 or GL_2
#
# Output: Returns true if the group is SL_2,
# false if GL_2
#
local x, Dx, f;
Dx := dom[1];
x := dom[2];
f := coeff( L, Dx );
rfldtest( f, x );
end:

138
o3np := proc( a, b, x )
#
#
#
#
# Takes as input the polynomials a and b in the
# indeterminate x, with algebraic number coefficients.
#
#
# Computes the Galois group of Dx^3 + a*Dx + b over the field
# of rational functions with algebraic number coefficients.
#
#
# Returns as output a list of the form
#
# [ U, P, Conj ].
#
# where U is the name of a unipotent group, P is the name of a
# reductive group, and the conjugation action of P on U is
# described by Conj.
#
#
# A nontrivial unipotent group U is one of the following:
#
# "U3", "C^2", "C", "0".
#
#
# A reductive group P is one of the following:
#
# "SL3", "PSL2", "GL2", "SL2", "C*^2" (i.e., C* x C*), "C*", "1".
#
#
#
# The conjugation action Conj for a semidirect product
# is represented in one of the following ways, depending on U and P:
#
# *** If U = 0, then Conj = "0"
#
# *** C^2 by SL2 or GL2:
# Conj = "matrix_vector" or "vector_transpose_matrix_inverse"
#
# *** C by C*:
# Conj = d, where t.u = t^d * u for t in C*, u in C
#
# *** C by C*^2:
# Conj = [d1,d2], where (t1,t2).u = t1^d1 * t2^d2 * u

139
# for t1, t2 in C*, u in C
#
# *** C^2 by C*:
# Conj = [d1,d2], where t.(u1,u2) = (t^d1 * u1, t^d2 * u2)
# for t in C*, u1, u2 in C
#
# *** C^2 by C*^2:
# Conj = [ [d1,d2], [e1,e2] ] where
#
# (t1,t2).(u,v) = ( t1^d1 * t2^d2 * u, t1^e1 * t2^e2 * v )
#
# for t1, t2 in C*, u, v in C
#
# *** U3 by C*:
# Conj. = [d1, d2], where C* embeds in SL3 via
# t |--> diag( t^d1, t^d2, t^(-d1-d2) ) and
# U3 is the group of upper triangular matrices in SL3 with
# 1s along the diagonal.
#
# *** U3 by C*^2:
# Conj. = "standard". In this case, the group is
# conjugate to T3 intersect SL3,
# the group of upper triangular matrices in SL3.
# Thus there is only one possible conjugation action.
#
#
local Dx, dom, L, Ladj, Lfactors, rlist, r1, r2, r3, L1, L2,
L2sharp, Ltest, n1, n2, Ls2, Ls2factors,
t, GredList, f, g, s, Ltemp, Ltemp1, elist;
dom := [Dx, x];
L := Dx^3 + a*Dx + b;
Lfactors := DFactor( L, dom );
n1 := numES( L, dom );
n2 := numES( Ladj, dom );
if ( n1 = 0 ) then
if ( n2 = 0 ) then
#
# n1 = n2 = 0
#
Ls2 := symmetric_power(L, 2, dom);
if ( degree( Ls2, Dx ) = 5 ) then
return [ "0", "PSL2", "0" ] ;
fi;
Ls2factors := DFactor( Ls2, dom );
if ( nops( Ls2factors ) > 1 ) then
return [ "0", "PSL2", "0" ] ;
else
return [ "0", "SL3", "0" ] ;

140
fi;
elif ( n2 = 1 ) then
#
# n1 = 0, n2 = 1
#
L2 := Lfactors[2];
if ( sl2test( L2, dom ) ) then
return [ "C^2", "SL2", "matrix_vector" ] ;
else
return [ "C^2", "GL2", "matrix_vector" ] ;
fi;
else
error "for n1 = 0, unexpected n2: %1", n2;
fi;
elif ( n1 = 1 ) then
if ( n2 = 0 ) then
#
# n1 = 1, n2 = 0
#
L2 := Lfactors[1];
if ( sl2test( L2, dom ) ) then
return [ "C^2", "SL2", "matrix_vector" ] ;
else
return [ "C^2", "GL2", "vector_transpose_matrix_inverse" ] ;
fi;
elif ( n2 = 1 ) then
#
# n1 = n2 = 1
#
if ( nops( Lfactors ) = 2 ) then
#
# L is a LCLM of an irreducible 2nd-order and a
# 1st-order operator
#
L1 := Lfactors[1];
L2 := Lfactors[2];
#
# Define Ltest to be the second-order factor of L,
# then apply sl2test. NOTE: Could also seek rational
# solutions of the first-order factor of L
#
#
if ( degree( L1, Dx ) = 2 ) then
Ltest := L1;
else
Ltest := L2;
fi;
if ( sl2test( Ltest, dom ) ) then
return [ "0", "SL2", "0" ] ;

141
else
return [ "0", "GL2", "0" ] ;
fi;
else
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