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###############################################################################
##
#F ppowerpolyloc.gi The SymbCompCC package Dörte Feichtenschlager
##
###############################################################################
##
## For a/b where a and b are p-power-polys elements and b is not divisible by
## p, the underlying prime.
##
###############################################################################
##
#M PPPL_Check( elm_in )
##
## Input: p-power-poly-loc element elm_in = [ elm1, elm2 ] where elm1 and
## elm2 are p-power-poly elements
##
## Output: [ elm1, elm2 ], where elm1/elm2 is probably reduced, or an Error
## occurs if the input does not represent a p-power-poly-loc element.
##
InstallGlobalFunction( PPPL_Check,
function( elm_in )
local elm1, elm2, p, Zero0, One1, test, elm, list, gcd_d, gcd_n, gcd,
ec1_num, ec2_num, i;
p := elm_in[1][1];
if p <> elm_in[2][1] then
Error("Wrong input. The underlying primes are different.");
fi;
Zero0 := PPP_ZeroNC( elm_in[1] );
One1 := PPP_OneNC( elm_in[1] );
if PPP_PadicValue( elm_in[2] ) <> [ 0, 0 ] then
Error("Wrong input. The denominator shall not be divisible by the underlying prime. ");
fi;
if PPP_Smaller( elm_in[2], Zero0 ) then
elm1 := PPP_AdditiveInverse( elm_in[1] );
elm2 := PPP_AdditiveInverse( elm_in[2] );
else
elm1 := elm_in[1];
elm2 := elm_in[2];
fi;
list := PPP_QuotientRemainder( elm1, elm2 );
if PPP_Equal( list[2], Zero0 ) then
elm1 := list[1];
elm2 := One1;
fi;
## check if we can cancel
if not PPP_Equal( elm1, Zero0 ) then
## take the numerators of the coefficients
ec1_num := [];
for i in [1..Length( elm1[2] )] do
ec1_num[i] := NumeratorRat( elm1[2][i] );
od;
ec2_num := [];
for i in [1..Length( elm2[2] )] do
ec2_num[i] := NumeratorRat( elm2[2][i] );
od;
## check if the gcds are non trivial
gcd_n := Gcd( ec1_num );
gcd_d := Gcd( ec2_num );
if gcd_n <> 1 and gcd_d <> 1 then
gcd := Gcd( gcd_d, gcd_n );
elm1[2] := elm1[2] / gcd;
elm2[2] := elm2[2] / gcd;
fi;
else elm2 := One1;
fi;
return [elm1, elm2];
end);
###############################################################################
##
#M PPPL_CheckNC( elm_in )
##
## Input: p-power-poly-loc element elm_in = [ elm1, elm2 ] where elm1 and
## elm2 are p-power-poly elements
##
## Output: [ elm1, elm2 ] or an Error occurs if the input does not represent
## a p-power-poly-loc element.
##
InstallGlobalFunction( PPPL_CheckNC,
function( elm_in )
local elm1, elm2, p, Zero0, One1, test, elm, list, gcd_d, gcd_n, gcd,
ec1_num, ec2_num, i;
p := elm_in[1][1];
if p <> elm_in[2][1] then
Error("Wrong input. The underlying primes are different.");
fi;
Zero0 := PPP_ZeroNC( elm_in[1] );
One1 := PPP_OneNC( elm_in[1] );
if PPP_PadicValue( elm_in[2] ) <> [ 0, 0 ] then
Error("Wrong input. The denominator shall not be divisible by the underlying prime. ");
fi;
if PPP_Smaller( elm_in[2], Zero0 ) then
elm1 := PPP_AdditiveInverse( elm_in[1] );
elm2 := PPP_AdditiveInverse( elm_in[2] );
else
elm1 := elm_in[1];
elm2 := elm_in[2];
fi;
## check if we can cancel
if not PPP_Equal( elm1, Zero0 ) then
## take the numerators of the coefficients
ec1_num := [];
for i in [1..Length( elm1[2] )] do
ec1_num[i] := NumeratorRat( elm1[2][i] );
od;
ec2_num := [];
for i in [1..Length( elm2[2] )] do
ec2_num[i] := NumeratorRat( elm2[2][i] );
od;
## check if the gcds are non trivial
gcd_n := Gcd( ec1_num );
gcd_d := Gcd( ec2_num );
if gcd_n <> 1 and gcd_d <> 1 then
gcd := Gcd( gcd_d, gcd_n );
elm1[2] := elm1[2] / gcd;
elm2[2] := elm2[2] / gcd;
fi;
else elm2 := One1;
fi;
return [elm1, elm2];
end);
###############################################################################
##
#M PPPL_Print( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the function prints elm in an easy to read way.
##
InstallGlobalFunction( PPPL_Print,
function( elm )
Print( " (" );
PPP_Print( elm[1] );
Print( ") / (" );
PPP_Print( elm[2] );
Print( ") " );
end);
###############################################################################
##
#M PPPL_PrintMat( obj )
##
## Input: a matrix obj with p-power-poly-loc entries.
##
## Output: the function prints obj in an easy to read way.
##
InstallGlobalFunction( PPPL_PrintMat,
function( obj )
local i, j;
Print( "[" );
for i in [1..Length( obj )] do
Print( "[ " );
PPPL_Print( obj[i][1] );
for j in [2..Length( obj[i] )] do
Print( ", " );
PPPL_Print( obj[i][j] );
od;
Print( "] " );
if i <> Length( obj ) then
Print( ",\n" );
fi;
od;
Print( "]\n" );
end);
###############################################################################
##
#M PPPL_PrintRow( obj )
##
## Input: a row of p-power-poly-loc elements.
##
## Output: the function prints obj in an easy to read way.
##
InstallGlobalFunction( PPPL_PrintRow,
function( obj )
local i, j;
Print( "[ " );
if IsInt( obj[1] ) then
Print( obj[1] );
else PPPL_Print( obj[1] );
fi;
for i in [2..Length( obj )] do
Print( ", " );
if IsInt( obj[i] ) then
Print( obj[i] );
else PPPL_Print( obj[i] );
fi;
od;
Print( " ]\n" );
end);
#############################################################################
##
#M PPPL_Equal( elm1, elm2 )
##
## Input: p-power-poly-loc elements elm1 and elm2.
##
## Output: a boolean, indicating whether elm1 is equal to elm2.
##
InstallGlobalFunction( PPPL_Equal,
function( elm1, elm2 )
local num1, num2, denom1, denom2;
num1 := elm1[1];
num2 := elm2[1];
denom1 := elm1[2];
denom2 := elm2[2];
if num1[1] <> num2[1] then
return false;
fi;
return PPP_Equal( PPP_Mult( num1, denom2 ), PPP_Mult( num2, denom1 ) );
end);
#############################################################################
##
#M PPPL_Smaller( elm1, elm2 )
##
## Input: p-power-poly-loc elements
##
## Output: a boolean, indicating whether elm1 is smaller than elm2.
##
InstallGlobalFunction( PPPL_Smaller,
function( elm1, elm2 )
local num1, num2, denom1, denom2;
num1 := elm1[1];
num2 := elm2[1];
denom1 := elm1[2];
denom2 := elm2[2];
if PPPL_Equal( elm1, elm2 ) then return false; fi;
if num1[1] < num2[1] then
return true;
elif num1[1] > num2[1] then
return false;
fi;
return PPP_Smaller( PPP_Mult( num1, denom2 ), PPP_Mult( num2, denom1 ) );
end);
#############################################################################
##
#M PPPL_Greater( elm1, elm2 )
##
## Input: p-power-poly-loc elements
##
## Output: a boolean, indicating whether elm1 is greater than elm2.
##
InstallGlobalFunction( PPPL_Greater,
function( elm1, elm2 )
local num1, num2, denom1, denom2;
num1 := elm1[1];
num2 := elm2[1];
denom1 := elm1[2];
denom2 := elm2[2];
if PPPL_Equal( elm1, elm2 ) then return false; fi;
if num1[1] > num2[1] then
return true;
elif num1[1] < num2[1] then
return false;
fi;
return PPP_Greater( PPP_Mult( num1, denom2 ), PPP_Mult( num2, denom1 ) );
end);
#############################################################################
##
#M PPPL_Add( elm1, elm2 )
##
## Input: p-power-poly-loc elements elm1 and elm2.
##
## Output: the sum of elm1 and elm2.
##
InstallGlobalFunction( PPPL_Add,
function( elm1, elm2 )
local num1, num2, denom1, denom2, num, Zero0, One1, denom, list, elm, p,
Zero0Loc;
## catch trivial case
if Length( elm1 ) = 4 and Length( elm2 ) = 4 then
return [ PPP_Add( elm1, elm2 ), PPP_OneNC( elm1 ) ];
fi;
if IsInt( elm1 ) and IsInt( elm2 ) then
Error( "No underlying prime is known." );
fi;
## check input
if Length( elm1 ) = 4 then
One1 := PPP_OneNC( elm1 );
elm1 := [ elm1, One1 ];
elif IsInt( elm1 ) then
if Length( elm2 ) = 4 then
p := elm2[1];
One1 := PPP_OneNC( elm2 );
else p := elm2[1][1];
One1 := PPP_OneNC( elm2[1] );
fi;
elm1 := [ Int2PPowerPoly( p, elm1 ), One1 ];
fi;
if Length( elm2 ) = 4 then
One1 := PPP_OneNC( elm2 );
elm2 := [ elm2, One1 ];
elif IsInt( elm2 ) then
if Length( elm1 ) = 4 then
p := elm1[1];
One1 := PPP_OneNC( elm1 );
else p := elm1[1][1];
One1 := PPP_OneNC( elm1[1] );
fi;
elm2 := [ Int2PPowerPoly( p, elm2 ), One1 ];
fi;
## initialize
num1 := StructuralCopy( elm1[1] );
num2 := StructuralCopy( elm2[1] );
denom1 := StructuralCopy( elm1[2] );
denom2 := StructuralCopy( elm2[2] );
## check
if num1[1] <> num2[1] then
Error( "The underlying primes are different." );
fi;
Zero0 := PPP_ZeroNC( num1 );
One1 := PPP_OneNC( denom1 );
Zero0Loc := PPPL_ZeroNC( elm1 );
## same denominator, so only add numerators
if PPP_Equal( denom1, denom2 ) then
num := PPP_Add( num1, num2 );
denom := denom1;
if PPP_Equal( num, Zero0 ) then
return Zero0Loc;
else
## check if we can cancel
list := PPP_QuotientRemainder( num, denom );
if PPP_Equal( list[2], Zero0 ) then
num := list[1];
denom := One1;
fi;
return PPPL_CheckNC( [ num, denom ] );
fi;
else
## fractional arithmetic
num := PPP_Add( PPP_Mult( denom2, num1 ), PPP_Mult( denom1, num2 ) );
denom := PPP_Mult( denom1, denom2 );
if PPP_Equal( num, Zero0 ) then
return Zero0Loc;
else
## check if we can cancel
list := PPP_QuotientRemainder( num, denom );
if PPP_Equal( list[2], Zero0 ) then
num := list[1];
denom := One1;
fi;
return PPPL_CheckNC( [ num, denom ] );
fi;
fi;
end);
#############################################################################
##
#M PPPL_Subtract( elm1, elm2 )
##
## Input: p-power-poly-loc elements elm1 and elm2.
##
## Output: the difference elm1 - elm2.
##
InstallGlobalFunction( PPPL_Subtract,
function( elm1, elm2 )
return PPPL_Add( elm1, PPPL_AdditiveInverse( elm2 ) );
end);
#############################################################################
##
#M PPPL_Mult( elm1, elm2 )
##
## Input: p-power-poly-loc elements elm1 and elm2.
##
## Output: the product of elm1 and elm2.
##
InstallGlobalFunction( PPPL_Mult,
function( elm1, elm2 )
local num1, num2, denom1, denom2, num, denom, elm, Zero0, One1, list_1,
list_2, p;
## catch trivial case
if Length( elm1 ) = 4 and Length( elm2 ) = 4 then
return [ PPP_Mult( elm1, elm2 ), PPP_OneNC( elm1 ) ];
fi;
if IsInt( elm1 ) and IsInt( elm2 ) then
Error( "No underlying prime is known." );
fi;
## check input
if Length( elm1 ) = 4 then
One1 := PPP_OneNC( elm1 );
elm1 := [ elm1, One1 ];
elif IsInt( elm1 ) then
if Length( elm2 ) = 4 then
p := elm2[1];
One1 := PPP_OneNC( elm2 );
else p := elm2[1][1];
One1 := PPP_OneNC( elm2[1] );
fi;
elm1 := [ Int2PPowerPoly( p, elm1 ), One1 ];
fi;
if Length( elm2 ) = 4 then
One1 := PPP_OneNC( elm2 );
elm2 := [ elm2, One1 ];
elif IsInt( elm2 ) then
if Length( elm1 ) = 4 then
p := elm1[1];
One1 := PPP_OneNC( elm1 );
else p := elm1[1][1];
One1 := PPP_OneNC( elm1[1] );
fi;
elm2 := [ Int2PPowerPoly( p, elm2 ), One1 ];
fi;
## initialize
num1 := StructuralCopy( elm1[1] );
num2 := StructuralCopy( elm2[1] );
denom1 := StructuralCopy( elm1[2] );
denom2 := StructuralCopy( elm2[2] );
## check
if num1[1] <> num2[1] then return fail; fi;
Zero0 := PPP_ZeroNC( num1 );
One1 := PPP_OneNC( denom1 );
## check if we can cancel
list_1 := PPP_QuotientRemainder( num1, denom2 );
list_2 := PPP_QuotientRemainder( num2, denom1 );
if PPP_Equal( list_1[2], Zero0 ) then
num := list_1[1];
denom := One1;
else
num := num1;
denom := denom2;
fi;
if PPP_Equal( list_2[2], Zero0 ) then
num := PPP_Mult( num, list_2[1] );
else
num := PPP_Mult( num, num2 );
denom := PPP_Mult( denom, denom1 );
fi;
return PPPL_CheckNC( [ num, denom ] );
end);
###############################################################################
##
## PPPL_OneNC( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the corresponding one p-power-poly-loc element.
##
InstallGlobalFunction( PPPL_OneNC,
function( elm )
local One1;
One1 := PPP_OneNC( elm[1] );
return [ One1, One1 ];
end);
###############################################################################
##
## PPPL_ZeroNC( elm )
##
## Input: a p-power-poly-loc element.
##
## Output: the corresponding zero p-power-poly-loc element.
##
InstallGlobalFunction( PPPL_ZeroNC,
function( elm )
local Zero0, One1;
Zero0 := PPP_ZeroNC( elm[1] );
One1 := PPP_OneNC( elm[1] );
return [ Zero0, One1 ];
end);
#############################################################################
##
#M PPPL_AdditiveInverse( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the additive inverse of elm.
##
InstallGlobalFunction( PPPL_AdditiveInverse,
function( elm )
local num, denom;
num := elm[1];
denom := elm[2];
return [ PPP_AdditiveInverse( num ), denom ];
end);
###############################################################################
##
#M PPPL_AbsValue( elm )
##
## Input: a p-power-poly-loc element.
##
## Output: the absolute value of elm.
##
InstallGlobalFunction( PPPL_AbsValue,
function( elm )
local num, denom, p, One1, Zero0Quot, coeffs;
## initialize
num := elm[1];
## make sure lead-coefficient of numerator is positive
coeffs := num[2];
if coeffs[Length( coeffs) ] < 0 then
return PPPL_AdditiveInverse( elm );
else return elm;
fi;
end);
###############################################################################
##
#M PPPL_PadicValue( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the p-adic value of elm, which is the p-adic value of the numerator.
##
InstallGlobalFunction( PPPL_PadicValue,
function( elm )
if not IsPDivisiblePPP( elm[2] ) then return fail; fi;
return PPP_PadicValue( elm[1] );
end);
###############################################################################
##
#M PPartPolyLoc( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the p-part of elm as a p-power-poly-loc element.
##
InstallGlobalFunction( PPartPolyLoc,
function( elm )
local num, One1, ppart_num;
if not IsPDivisiblePPP( elm[2] ) then return fail; fi;
num := elm[1];
One1 := One( num );
ppart_num := PPartPoly( num );
return [ ppart_num, One1 ];
end);
###############################################################################
##
#M PPrimePartPolyLoc( elm )
##
## Input: a p-power-poly-loc element elm.
##
## Output: the p-prime part of elm as a p-power-poly-loc element.
##
InstallGlobalFunction( PPrimePartPolyLoc,
function( elm )
local pprimepart_num;
if not IsPDivisiblePPP( elm[2] ) then return fail; fi;
pprimepart_num := PPrimePartPoly( elm[1] );
return [ pprimepart_num, elm[2] ];
end);
###############################################################################
##
## PPowerPolyMat2PPowerPolyLocMat( M )
##
## Inpput: a matrix M with p-power-poly entries.
##
## Output: a matrix where the entries are the corresponding p-power-poly-loc
## elements.
##
InstallMethod( PPowerPolyMat2PPowerPolyLocMat, [ IsList ],
function( M )
local One1, M_Q, i, j;
One1 := Int2PPowerPoly( M[1][1][1], 1 );
M_Q := [];
for i in [1..Length(M)] do
M_Q[i] := [];
for j in [1..Length(M[i])] do
M_Q[i][j] := [ M[i][j], One1 ];
od;
od;
return M_Q;
end);
###############################################################################
##
#M PPPL_MatrixMult( A, B )
##
## Input: p-power-poly-loc matrices A and B.
##
## Ouput: a matrix which is the product of A and B.
##
InstallGlobalFunction( PPPL_MatrixMult,
function( A, B )
local C, n_A, m_A, n_B, m_B, i, j, k, Zero0Loc;
n_A := Length( A );
m_A := Length( A[1] );
n_B := Length( B );
m_B := Length( B[1] );
if n_B <> m_A then
Error( "The dimensions of the matrices do not fit together." );
fi;
Zero0Loc := PPPL_ZeroNC( A[1][1] );
C := [];
for i in [1..n_A] do
C[i] := [];
for j in [1..m_B] do
C[i][j] := Zero0Loc;
for k in [1..n_B] do
C[i][j] := PPPL_Add( C[i][j], PPPL_Mult( A[i][k], B[k][j] ) );
od;
od;
od;
return C;
end);
#E ppowerpolyloc.gi . . . . . . . . . . . . . . . . . . . . . . . . ends here
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