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# SPDX-License-Identifier: GPL-2.0-or-later
# RingsForHomalg: Dictionaries of external rings
#
# Implementations
#
## Implementations for the rings provided by MAGMA.
####################################
#
# global variables:
#
####################################
BindGlobal( "CommonHomalgTableForMAGMATools",
rec(
Zero := HomalgExternalRingElement( R -> homalgSendBlocking( [ "Zero(", R, ")" ], "Zero" ), "MAGMA", IsZero ),
One := HomalgExternalRingElement( R -> homalgSendBlocking( [ "One(", R, ")" ], "One" ), "MAGMA", IsOne ),
MinusOne := HomalgExternalRingElement( R -> homalgSendBlocking( [ "-One(", R, ")" ], "MinusOne" ), "MAGMA", IsMinusOne ),
RingElement := R -> r -> homalgSendBlocking( [ R, "!(", r, ")" ], "define" ),
IsZero := r -> homalgSendBlocking( [ "IsZero(", r, ")" ] , "need_output", "IsZero" ) = "true",
IsOne := r -> homalgSendBlocking( [ "IsOne(", r, ")" ] , "need_output", "IsOne" ) = "true",
Minus :=
function( a, b )
return homalgSendBlocking( [ a, "-(", b, ")" ], "Minus" );
end,
DivideByUnit :=
function( a, u )
return homalgSendBlocking( [ "(", a, ")/(", u, ")" ], "DivideByUnit" );
end,
IsUnit :=
function( R, u )
return homalgSendBlocking( [ "IsUnit(", R, "!", u, ")" ], "need_output", "IsUnit" ) = "true";
end,
Sum :=
function( a, b )
return homalgSendBlocking( [ a, "+(", b, ")" ], "Sum" );
end,
Product :=
function( a, b )
return homalgSendBlocking( [ "(", a, ")*(", b, ")" ], "Product" );
end,
Gcd :=
function( a, b )
return homalgSendBlocking( [ "GreatestCommonDivisor(", a, b, ")" ], "CancelGcd" );
end,
CancelGcd :=
function( a, b )
local a_g, b_g;
homalgSendBlocking( [ "g:=GreatestCommonDivisor(", a, b, ")" ], "need_command", "Gcd" );
a_g := homalgSendBlocking( [ "(", a, ") div g" ], "CancelGcd" );
b_g := homalgSendBlocking( [ "(", b, ") div g" ], "CancelGcd" );
return [ a_g, b_g ];
end,
# CopyElement :=
# function( r, R )
# return homalgSendBlocking( [ R, "!", r ], HomalgRing( r ), "CopyElement" );
# end,
ShallowCopy := C -> homalgSendBlocking( [ C ], "CopyMatrix" ),
CopyMatrix :=
function( C, R )
local S;
S := HomalgRing( C );
if HasRelativeIndeterminatesOfPolynomialRing( R ) and
HasCoefficientsRing( S ) and
HasIsFieldForHomalg( CoefficientsRing( S ) ) and IsFieldForHomalg( CoefficientsRing( S ) ) then
return Eval( ConvertHomalgMatrixViaFile( C, R ) );
fi;
return homalgSendBlocking( [ "imap(", C, R, ")" ], "CopyMatrix" );
end,
ZeroMatrix :=
function( C )
return homalgSendBlocking( [ "ZeroMatrix(", HomalgRing( C ), NumberRows( C ), NumberColumns( C ), ")" ], "ZeroMatrix" );
end,
IdentityMatrix :=
function( C )
return homalgSendBlocking( [ "ScalarMatrix(", HomalgRing( C ), NumberRows( C ), ",1)" ], "IdentityMatrix" );
end,
AreEqualMatrices :=
function( A, B )
return homalgSendBlocking( [ A, " eq ", B ] , "need_output", "AreEqualMatrices" ) = "true";
end,
Involution := M -> homalgSendBlocking( [ "Transpose(", M, ")" ], "Involution" ),
TransposedMatrix := M -> homalgSendBlocking( [ "Transpose(", M, ")" ], "TransposedMatrix" ),
CertainRows :=
function( M, plist )
return homalgSendBlocking( [ "Matrix(", M, "[ \\", plist, "])" ], "CertainRows" );
end,
CertainColumns :=
function( M, plist )
return homalgSendBlocking( [ "Transpose(Matrix(Transpose(", M, ")[ \\",plist, "]))" ], "CertainColumns" );
end,
UnionOfRowsPair :=
function( A, B )
return homalgSendBlocking( [ "VerticalJoin(", A, B, ")" ], "UnionOfRows" );
end,
UnionOfColumnsPair :=
function( A, B )
return homalgSendBlocking( [ "HorizontalJoin(", A, B, ")" ], "UnionOfColumns" );
end,
DiagMat :=
function( e )
local f;
f := Concatenation( [ "DiagonalJoin(<" ], e, [ ">)" ] );
return homalgSendBlocking( f, "DiagMat" );
end,
KroneckerMat :=
function( A, B )
return homalgSendBlocking( [ "TensorProduct(", A, B, ")" ], "KroneckerMat" );
end,
MulMat :=
function( a, A )
return homalgSendBlocking( [ "(", a, ")*", A ], "MulMat" );
end,
MulMatRight :=
function( A, a )
return homalgSendBlocking( [ A, "*(", a, ")" ], "MulMatRight" );
end,
AddMat :=
function( A, B )
return homalgSendBlocking( [ A, "+", B ], "AddMat" );
end,
SubMat :=
function( A, B )
return homalgSendBlocking( [ A, "-", B ], "SubMat" );
end,
Compose :=
function( A, B )
return homalgSendBlocking( [ A, "*", B ], "Compose" );
end,
NumberRows :=
function( C )
return StringToInt( homalgSendBlocking( [ "NumberOfRows(", C, ")" ], "need_output", "NumberRows" ) );
end,
NumberColumns :=
function( C )
return StringToInt( homalgSendBlocking( [ "NumberOfColumns(", C, ")" ], "need_output", "NumberColumns" ) );
end,
Determinant :=
function( C )
return homalgSendBlocking( [ "Determinant(", C, ")" ], "Determinant" );
end,
IsZeroMatrix :=
function( M )
return homalgSendBlocking( [ "IsZero(", M, ")" ] , "need_output", "IsZeroMatrix" ) = "true";
end,
IsIdentityMatrix :=
function( M )
return homalgSendBlocking( [ "IsOne(", M, ")" ] , "need_output", "IsIdentityMatrix" ) = "true";
end,
IsDiagonalMatrix :=
function( M )
return homalgSendBlocking( [ "IsDiagonalMatrix(", M, ")" ] , "need_output", "IsDiagonalMatrix" ) = "true";
end,
ZeroRows :=
function( C )
local list_string;
list_string := homalgSendBlocking( [ "ZeroRows(", C, ")" ], "need_output", "ZeroRows" );
return StringToIntList( list_string );
end,
ZeroColumns :=
function( C )
local list_string;
list_string := homalgSendBlocking( [ "ZeroColumns(", C, ")" ], "need_output", "ZeroColumns" );
return StringToIntList( list_string );
end,
GetColumnIndependentUnitPositions :=
function( M, pos_list )
return StringToDoubleIntList( homalgSendBlocking( [ "GetColumnIndependentUnitPositions(", M, pos_list, ")" ], "need_output", "GetColumnIndependentUnitPositions" ) );
end,
GetRowIndependentUnitPositions :=
function( M, pos_list )
return StringToDoubleIntList( homalgSendBlocking( [ "GetRowIndependentUnitPositions(", M, pos_list, ")" ], "need_output", "GetRowIndependentUnitPositions" ) );
end,
GetUnitPosition :=
function( M, pos_list )
local list_string;
list_string := homalgSendBlocking( [ "GetUnitPosition(", M, pos_list, ")" ], "need_output", "GetUnitPosition" );
if list_string = "fail" then
return fail;
else
return StringToIntList( list_string );
fi;
end,
PositionOfFirstNonZeroEntryPerRow :=
function( M )
local L;
L := homalgSendBlocking( [ "PositionOfFirstNonZeroEntryPerRow( ", M, " )" ], "need_output", "PositionOfFirstNonZeroEntryPerRow" );
L := StringToIntList( L );
if Length( L ) = 1 then
return ListWithIdenticalEntries( NumberRows( M ), L[1] );
fi;
return L;
end,
PositionOfFirstNonZeroEntryPerColumn :=
function( M )
local L;
L := homalgSendBlocking( [ "PositionOfFirstNonZeroEntryPerColumn( ", M, " )" ], "need_output", "PositionOfFirstNonZeroEntryPerColumn" );
L := StringToIntList( L );
if Length( L ) = 1 then
return ListWithIdenticalEntries( NumberColumns( M ), L[1] );
fi;
return L;
end,
DivideRowByUnit :=
function( M, i, u, j )
homalgSendBlocking( [ "DivideRowByUnit(~", M, i, u, j, ")" ], "need_command", "DivideRowByUnit" );
end,
DivideColumnByUnit :=
function( M, j, u, i )
homalgSendBlocking( [ "DivideColumnByUnit(~", M, j, u, i, ")" ], "need_command", "DivideColumnByUnit" );
end,
CopyRowToIdentityMatrix :=
function( M, i, L, j )
local l;
l := Length( L );
if l > 1 and ForAll( L, IsHomalgMatrix ) then
homalgSendBlocking( [ "CopyRowToIdentityMatrix2(", M, i, ",~", L[1], ",~", L[2], j, ")" ], "need_command", "CopyRowToIdentityMatrix" );
elif l > 0 and IsHomalgMatrix( L[1] ) then
homalgSendBlocking( [ "CopyRowToIdentityMatrix(", M, i, ",~", L[1], j, -1, ")" ], "need_command", "CopyRowToIdentityMatrix" );
elif l > 1 and IsHomalgMatrix( L[2] ) then
homalgSendBlocking( [ "CopyRowToIdentityMatrix(", M, i, ",~", L[2], j, 1, ")" ], "need_command", "CopyRowToIdentityMatrix" );
fi;
end,
CopyColumnToIdentityMatrix :=
function( M, j, L, i )
local l;
l := Length( L );
if l > 1 and ForAll( L, IsHomalgMatrix ) then
homalgSendBlocking( [ "CopyColumnToIdentityMatrix2(", M, j, ",~", L[1], ",~", L[2], i, ")" ], "need_command", "CopyColumnToIdentityMatrix" );
elif l > 0 and IsHomalgMatrix( L[1] ) then
homalgSendBlocking( [ "CopyColumnToIdentityMatrix(", M, j, ",~", L[1], i, -1, ")" ], "need_command", "CopyColumnToIdentityMatrix" );
elif l > 1 and IsHomalgMatrix( L[2] ) then
homalgSendBlocking( [ "CopyColumnToIdentityMatrix(", M, j, ",~", L[2], i, 1, ")" ], "need_command", "CopyColumnToIdentityMatrix" );
fi;
end,
SetColumnToZero :=
function( M, i, j )
homalgSendBlocking( [ "SetColumnToZero(~", M, i, j, ")" ], "need_command", "SetColumnToZero" );
end,
ConvertRowToMatrix :=
function( M, r, c )
return homalgSendBlocking( [ "ConvertRowToMatrix(", M, r, c, HomalgRing( M ), ")" ], "ConvertRowToMatrix" );
end,
GetCleanRowsPositions :=
function( M, clean_columns )
local list_string;
list_string := homalgSendBlocking( [ "GetCleanRowsPositions(", M, clean_columns, ")" ], "need_output", "GetCleanRowsPositions" );
return StringToIntList( list_string );
end,
AffineDimensionOfIdeal :=
function( mat )
local R, v;
R := HomalgRing ( mat );
v := homalgStream( R )!.variable_name;
mat := EntriesOfHomalgMatrix( mat );
return Int( homalgSendBlocking( [ v, "d := Dimension(ideal<", R, "|", mat, ">); ", v, "d" ], "break_lists", "need_output", "AffineDimension" ) );
end,
## do not add CoefficientsOf(Unreduced)NumeratorOfHilbertPoincareSeries
## since MAGMA does not support Hilbert* for non-graded modules
CoefficientsOfUnreducedNumeratorOfWeightedHilbertPoincareSeries :=
function( mat, weights, degrees )
local R, v, hilb;
if Set( weights ) <> [ 1 ] then
Error( "the homalgTable entry CoefficientsOfUnreducedNumeratorOfWeightedHilbertPoincareSeries for MAGMA does not yet support weights\n" );
fi;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
hilb := homalgSendBlocking( [ v, "numer,", v, "ldeg:=", "HilbertNumerator(quo<GradedModule(", R, ",[", degrees, "])|RowSequence(", mat, ")>); Append(Coefficients(", v, "numer),-", v, "ldeg)" ], "break_lists", "need_output", "HilbertPoincareSeries" );
return StringToIntList( hilb );
end,
RadicalSubobject :=
function( mat )
local R;
R := HomalgRing( mat );
if not NumberColumns( mat ) = 1 then
Error( "only radical of one-column matrices is supported\n" );
fi;
mat := EntriesOfHomalgMatrix( mat );
return homalgSendBlocking( [ "Transpose(Matrix([GroebnerBasis(Radical(ideal<", R, "|", mat, ">))]))" ], "RadicalSubobject" );
end,
IsPrime :=
function( mat )
local R;
R := HomalgRing( mat );
mat := EntriesOfHomalgMatrix( mat );
return homalgSendBlocking( [ "IsPrime(ideal<", R, "|", mat, ">)" ], "need_output", "break_lists", "PrimaryDecomposition" ) = "true";
end,
PrimaryDecomposition :=
function( mat )
local R, v, c;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
mat := EntriesOfHomalgMatrix( mat );
homalgSendBlocking( [ v, "Q, ", v, "P := PrimaryDecomposition(ideal<", R, "|", mat, ">)" ], "need_command", "break_lists", "PrimaryDecomposition" );
homalgSendBlocking( [ v, "Q := [ GroebnerBasis( x ) : x in ", v, "Q ]" ], R, "need_command", "PrimaryDecomposition" );
homalgSendBlocking( [ v, "P := [ GroebnerBasis( x ) : x in ", v, "P ]" ], R, "need_command", "PrimaryDecomposition" );
c := Int( homalgSendBlocking( [ "#", v, "Q" ], "need_output", R, "PrimaryDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local primary, prime;
primary := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ primary, " := Matrix(", R, ", #(", v, "Q[", i, "]), 1, ", v, "Q[", i, "] )" ], "need_command", "PrimaryDecomposition" );
homalgSendBlocking( [ prime, " := Matrix(", R, ", #(", v, "P[", i, "]), 1, ", v, "P[", i, "] )" ], "need_command", "PrimaryDecomposition" );
return [ primary, prime ];
end
);
end,
RadicalDecomposition :=
function( mat )
local R, v, c;
R := HomalgRing( mat );
v := homalgStream( R )!.variable_name;
mat := EntriesOfHomalgMatrix( mat );
homalgSendBlocking( [ v, "P := RadicalDecomposition(ideal<", R, "|", mat, ">)" ], "need_command", "break_lists", "PrimaryDecomposition" );
homalgSendBlocking( [ v, "P := [ GroebnerBasis( x ) : x in ", v, "P ]" ], R, "need_command", "PrimaryDecomposition" );
c := Int( homalgSendBlocking( [ "#", v, "P" ], "need_output", R, "PrimaryDecomposition" ) );
return
List( [ 1 .. c ],
function( i )
local prime;
prime := HomalgVoidMatrix( "unkown_number_of_rows", 1, R );
homalgSendBlocking( [ prime, " := Matrix(", R, ", #(", v, "P[", i, "]), 1, ", v, "P[", i, "] )" ], "need_command", "PrimaryDecomposition" );
return prime;
end
);
end,
Eliminate :=
function( rel, indets, R )
local elim;
elim := Difference( Indeterminates( R ), indets );
return homalgSendBlocking( [ "Transpose(Matrix([GroebnerBasis(EliminationIdeal(ideal<", R, "|", rel, ">,{", elim, "}))]))" ], "break_lists", "Eliminate" );
end,
Coefficients :=
function( poly, var )
local R, y, vars, coeffs, d;
R := HomalgRing( poly );
if HasRelativeIndeterminatesOfPolynomialRing( R ) then
y := RelativeIndeterminatesOfPolynomialRing( R );
if y <> var then
Error( "the list of given variables does not coincide with the list of relative indeterminates\n" );
elif Length( y ) > 1 then
Error( "this table entry can only handle univariate polynomial rings over some base ring\n" );
fi;
y := y[1];
vars := homalgSendBlocking( [ "Reverse(MonomialsUnivariate(", poly, y, "))" ], R, "Coefficients" );
coeffs := homalgSendBlocking( [ "Reverse(Coefficients(", poly, y, "))" ], R, "Coefficients" );
else
vars := homalgSendBlocking( [ "Monomials(", poly, ")" ], R, "Coefficients" );
coeffs := homalgSendBlocking( [ "Coefficients(", poly, ")" ], R, "Coefficients" );
fi;
d := Int( homalgSendBlocking( [ "#", coeffs ], "need_output", R, "Coefficients" ) );
homalgSendBlocking( [ coeffs, ":=Matrix(", R, d, 1, coeffs, ")" ], "need_command", R, "Coefficients" );
coeffs := HomalgMatrix( coeffs, d, 1, R );
d := NonZeroRows( coeffs );
coeffs := Eval( CertainRows( coeffs, d ) );
homalgSendBlocking( [ vars, ":=Matrix(", R, Length( d ), 1, vars, ")" ], "need_command", R, "Coefficients" );
return [ vars, coeffs ];
end,
DegreeOfRingElement :=
function( r, R )
local y;
if HasRelativeIndeterminatesOfPolynomialRing( R ) then
y := RelativeIndeterminatesOfPolynomialRing( R );
if Length( y ) > 1 then
Error( "this table entry can only handle univariate polynomial rings over some base ring\n" );
fi;
y := y[1];
return Int( homalgSendBlocking( [ "Deg2(", r, R, y, ")" ], "need_output", "DegreeOfRingElement" ) );
fi;
return Int( homalgSendBlocking( [ "Deg(", r, R, ")" ], "need_output", "DegreeOfRingElement" ) );
end,
CoefficientsOfUnivariatePolynomial :=
function( r, var )
local R, y, coeffs, d;
R := HomalgRing( r );
if HasRelativeIndeterminatesOfPolynomialRing( R ) then
y := RelativeIndeterminatesOfPolynomialRing( R );
if y <> [ var ] then
Error( "the list of given variables does not coincide with the list of relative indeterminates\n" );
elif Length( y ) > 1 then
Error( "this table entry can only handle univariate polynomial rings over some base ring\n" );
fi;
y := y[1];
else
y := var;
fi;
coeffs := homalgSendBlocking( [ "Coefficients(", r, y, ")" ], R, "Coefficients" );
d := Int( homalgSendBlocking( [ "#", coeffs ], "need_output", R, "Coefficients" ) );
homalgSendBlocking( [ coeffs, ":=Matrix(", R, 1, d, coeffs, ")" ], "need_command", R, "Coefficients" );
return coeffs;
end,
LeadingIdeal :=
function( mat )
local R;
R := HomalgRing( mat );
return homalgSendBlocking( [ "Transpose(Matrix([GroebnerBasis(LeadingMonomialIdeal(ideal<", R, "|", EntriesOfHomalgMatrix( mat ), ">))]))" ], "break_lists", "LeadingModule" );
end,
MonomialMatrix :=
function( i, vars, R )
return homalgSendBlocking( [ "Matrix(1,MonomialsOfDegree(", R, i, ",{", R, ".i : i in [ 1 .. Rank(", R, ")]} diff {", vars, "}))" ], "break_lists", "MonomialMatrix" );
end,
Diff :=
function( D, N )
return homalgSendBlocking( [ "Diff(", D, N, ")" ], "Diff" );
end,
Pullback :=
function( phi, M )
if not IsBound( phi!.RingMap ) then
phi!.RingMap :=
homalgSendBlocking( [ "hom< ", Source( phi ), " -> ", Range( phi ), " | ", ImagesOfRingMap( phi ), " >" ], Range( phi ), "break_lists", "define" );
fi;
Eval( M ); ## the following line might reset Range( phi ) to HomalgRing( M ) in order to evaluate the (eventually lazy) matrix M -> error
return homalgSendBlocking( [ "ChangeRing(", M, phi!.RingMap, ")" ], Range( phi ), "Pullback" );
end,
Evaluate :=
function( p, L )
if Length( L ) > 2 then
Error( "MAGMA only supports Evaluate( p, var1, val1, var2, val2, ...)\n" );
fi;
return homalgSendBlocking( [ "Evaluate(", p, L, ")" ], "break_lists", "Evaluate" );
end,
NumeratorAndDenominatorOfPolynomial :=
function( p )
local R, numer, denom;
R := HomalgRing( p );
#numer := homalgSendBlocking( [ "Numerator(", p, ")" ], R, "Numerator" );
denom := homalgSendBlocking( [ "Denominator(", p, ")" ], R, "Numerator" );
#numer := HomalgExternalRingElement( numer, R );
denom := HomalgExternalRingElement( denom, R );
numer := p * denom;
return [ numer, denom ];
end,
)
);
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