def invariant_generators(self):
r"""
Return invariant ring generators.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where `G` in `GL(n,F)` is a finite matrix
group.
In the "good characteristic" case the polynomials returned
form a minimal generating set for the algebra of `G`-invariant
polynomials. In the "bad" case, the polynomials returned
are primary and secondary invariants, forming a not
necessarily minimal generating set for the algebra of
`G`-invariant polynomials.
ALGORITHM:
Wraps Singular's ``invariant_algebra_reynolds`` and ``invariant_ring``
in ``finvar.lib``.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7,
x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12,
x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = CyclotomicField(8)
sage: z = F.gen()
sage: a = z+1/z
sage: b = z^2
sage: MS = MatrixSpace(F,2,2)
sage: g1 = MS([[1/a, 1/a], [1/a, -1/a]])
sage: g2 = MS([[-b, 0], [0, b]])
sage: G=MatrixGroup([g1,g2])
sage: G.invariant_generators()
[x1^4 + 2*x1^2*x2^2 + x2^4,
x1^5*x2 - x1*x2^5,
x1^8 + 28/9*x1^6*x2^2 + 70/9*x1^4*x2^4 + 28/9*x1^2*x2^6 + x2^8]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- [Stu1993]_
- S. King, "Minimal Generating Sets of non-modular invariant
rings of finite groups", :arxiv:`math/0703035`.
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen()==1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F,'polynomial'): # we got an algebraic extension
if len(F.gens())>1:
raise NotImplementedError("can only deal with finite fields and (simple algebraic extensions of) the rationals")
FieldStr = '(%d,%s)'%(F.characteristic(),str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)'%(F.characteristic(),','.join([str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0]=='x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames='('+','.join((VarStr+str(i+1) for i in range(n)))+')'
R=singular.ring(FieldStr,VarNames,'dp')
if hasattr(F,'polynomial') and F.gen()!=1: # we have to define minpoly
singular.eval('minpoly = '+str(F.polynomial()).replace('x',str(F.gen())))
A = [singular.matrix(n,n,str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F,n,[VarStr+str(i) for i in range(1,n+1)])
if q == 0 or (q > 0 and self.cardinality()%q != 0):
from sage.all import Integer, Matrix
try:
elements = [ g.matrix() for g in self.list() ]
except (TypeError,ValueError):
elements
if elements is not None:
ReyName = 't'+singular._next_var_name()
singular.eval('matrix %s[%d][%d]'%(ReyName,self.cardinality(),n))
for i in range(1,self.cardinality()+1):
M = Matrix(F, elements[i-1])
D = [{} for foobar in range(self.degree())]
for x,y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)'
%(ReyName,i,row+1,ReyName,i,row+1, repr(t[1]),t[0]+1))
foobar = singular(ReyName)
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)'%(IRName,ReyName))
else:
ReyName = 't'+singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))'%(ReyName,Lgens))
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])'%(IRName,ReyName))
OUT = [singular.eval(IRName+'[1,%d]'%(j))
for j in range(1,1+singular('ncols('+IRName+')'))]
return [PR(gen) for gen in OUT]
if self.cardinality()%q == 0:
PName = 't'+singular._next_var_name()
SName = 't'+singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)'%(PName,SName,Lgens))
OUT = [
singular.eval(PName+'[1,%d]'%(j))
for j in range(1,1+singular('ncols('+PName+')'))
] + [
singular.eval(SName+'[1,%d]'%(j)) for j in range(2,1+singular('ncols('+SName+')'))
]
return [PR(gen) for gen in OUT]
def invariant_generators(self):
r"""
Return invariant ring generators.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where `G` in `GL(n,F)` is a finite matrix
group.
In the "good characteristic" case the polynomials returned
form a minimal generating set for the algebra of `G`-invariant
polynomials. In the "bad" case, the polynomials returned
are primary and secondary invariants, forming a not
necessarily minimal generating set for the algebra of
`G`-invariant polynomials.
ALGORITHM:
Wraps Singular's ``invariant_algebra_reynolds`` and ``invariant_ring``
in ``finvar.lib``.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7,
x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12,
x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = CyclotomicField(8)
sage: z = F.gen()
sage: a = z+1/z
sage: b = z^2
sage: MS = MatrixSpace(F,2,2)
sage: g1 = MS([[1/a, 1/a], [1/a, -1/a]])
sage: g2 = MS([[-b, 0], [0, b]])
sage: G=MatrixGroup([g1,g2])
sage: G.invariant_generators()
[x1^4 + 2*x1^2*x2^2 + x2^4,
x1^5*x2 - x1*x2^5,
x1^8 + 28/9*x1^6*x2^2 + 70/9*x1^4*x2^4 + 28/9*x1^2*x2^6 + x2^8]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- [Stu1993]_
- S. King, "Minimal Generating Sets of non-modular invariant
rings of finite groups", :arxiv:`math/0703035`.
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen() == 1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F, 'polynomial'): # we got an algebraic extension
if len(F.gens()) > 1:
raise NotImplementedError(
"can only deal with finite fields and (simple algebraic extensions of) the rationals"
)
FieldStr = '(%d,%s)' % (F.characteristic(), str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)' % (F.characteristic(), ','.join(
[str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0] == 'x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames = '(' + ','.join(
(VarStr + str(i + 1) for i in range(n))) + ')'
R = singular.ring(FieldStr, VarNames,
'dp') # this does have a side-effect
if hasattr(F,
'polynomial') and F.gen() != 1: # we have to define minpoly
singular.eval('minpoly = ' +
str(F.polynomial()).replace('x', str(F.gen())))
A = [singular.matrix(n, n, str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F, n, [VarStr + str(i) for i in range(1, n + 1)])
if q == 0 or (q > 0 and self.cardinality() % q != 0):
from sage.all import Matrix
try:
elements = [g.matrix() for g in self.list()]
except (TypeError, ValueError):
elements
if elements is not None:
ReyName = 't' + singular._next_var_name()
singular.eval('matrix %s[%d][%d]' %
(ReyName, self.cardinality(), n))
for i in range(1, self.cardinality() + 1):
M = Matrix(F, elements[i - 1])
D = [{} for foobar in range(self.degree())]
for x, y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)' %
(ReyName, i, row + 1, ReyName, i,
row + 1, repr(t[1]), t[0] + 1))
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)' %
(IRName, ReyName))
else:
ReyName = 't' + singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))' %
(ReyName, Lgens))
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])' %
(IRName, ReyName))
OUT = [
singular.eval(IRName + '[1,%d]' % (j))
for j in range(1, 1 + int(singular('ncols(' + IRName + ')')))
]
return [PR(gen) for gen in OUT]
if self.cardinality() % q == 0:
PName = 't' + singular._next_var_name()
SName = 't' + singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)' %
(PName, SName, Lgens))
OUT = [
singular.eval(PName + '[1,%d]' % (j))
for j in range(1, 1 + singular('ncols(' + PName + ')'))
] + [
singular.eval(SName + '[1,%d]' % (j))
for j in range(2, 1 + singular('ncols(' + SName + ')'))
]
return [PR(gen) for gen in OUT]
def invariant_generators(self):
"""
Wraps Singular's invariant_algebra_reynolds and invariant_ring
in finvar.lib, with help from Simon King and Martin Albrecht.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where G in GL(n,F) is a finite
matrix group.
In the "good characteristic" case the polynomials returned form a
minimal generating set for the algebra of G-invariant polynomials.
In the "bad" case, the polynomials returned are primary and
secondary invariants, forming a not necessarily minimal generating
set for the algebra of G-invariant polynomials.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7, x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12, x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[-1,1]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators() # long time (67s on sage.math, 2012)
[x1^20 + x1^16*x2^4 + x1^12*x2^8 + x1^8*x2^12 + x1^4*x2^16 + x2^20, x1^20*x2^4 + x1^16*x2^8 + x1^12*x2^12 + x1^8*x2^16 + x1^4*x2^20]
sage: F=CyclotomicField(8)
sage: z=F.gen()
sage: a=z+1/z
sage: b=z^2
sage: MS=MatrixSpace(F,2,2)
sage: g1=MS([[1/a,1/a],[1/a,-1/a]])
sage: g2=MS([[1,0],[0,b]])
sage: g3=MS([[b,0],[0,1]])
sage: G=MatrixGroup([g1,g2,g3])
sage: G.invariant_generators() # long time (12s on sage.math, 2011)
[x1^8 + 14*x1^4*x2^4 + x2^8,
x1^24 + 10626/1025*x1^20*x2^4 + 735471/1025*x1^16*x2^8 + 2704156/1025*x1^12*x2^12 + 735471/1025*x1^8*x2^16 + 10626/1025*x1^4*x2^20 + x2^24]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- B. Sturmfels, "Algorithms in invariant theory", Springer-Verlag, 1993.
- S. King, "Minimal Generating Sets of non-modular invariant rings of
finite groups", arXiv:math.AC/0703035
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen() == 1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F, 'polynomial'): # we got an algebraic extension
if len(F.gens()) > 1:
raise NotImplementedError, "can only deal with finite fields and (simple algebraic extensions of) the rationals"
FieldStr = '(%d,%s)' % (F.characteristic(), str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)' % (F.characteristic(), ','.join(
[str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0] == 'x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames = '(' + ','.join(
(VarStr + str(i + 1) for i in range(n))) + ')'
R = singular.ring(FieldStr, VarNames, 'dp')
if hasattr(F,
'polynomial') and F.gen() != 1: # we have to define minpoly
singular.eval('minpoly = ' +
str(F.polynomial()).replace('x', str(F.gen())))
A = [singular.matrix(n, n, str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F, n, [VarStr + str(i) for i in range(1, n + 1)])
if q == 0 or (q > 0 and self.cardinality() % q != 0):
from sage.all import Integer, Matrix
try:
gapself = gap(self)
# test whether the backwards transformation works as well:
for i in range(self.ngens()):
if Matrix(gapself.gen(i + 1), F) != self.gen(i).matrix():
raise ValueError
except (TypeError, ValueError):
gapself is None
if gapself is not None:
ReyName = 't' + singular._next_var_name()
singular.eval('matrix %s[%d][%d]' %
(ReyName, self.cardinality(), n))
En = gapself.Enumerator()
for i in range(1, self.cardinality() + 1):
M = Matrix(En[i], F)
D = [{} for foobar in range(self.degree())]
for x, y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)' %
(ReyName, i, row + 1, ReyName, i,
row + 1, repr(t[1]), t[0] + 1))
foobar = singular(ReyName)
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)' %
(IRName, ReyName))
else:
ReyName = 't' + singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))' %
(ReyName, Lgens))
IRName = 't' + singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])' %
(IRName, ReyName))
OUT = [
singular.eval(IRName + '[1,%d]' % (j))
for j in range(1, 1 + singular('ncols(' + IRName + ')'))
]
return [PR(gen) for gen in OUT]
if self.cardinality() % q == 0:
PName = 't' + singular._next_var_name()
SName = 't' + singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)' %
(PName, SName, Lgens))
OUT = [
singular.eval(PName + '[1,%d]' % (j))
for j in range(1, 1 + singular('ncols(' + PName + ')'))
] + [
singular.eval(SName + '[1,%d]' % (j))
for j in range(2, 1 + singular('ncols(' + SName + ')'))
]
return [PR(gen) for gen in OUT]
def invariant_generators(self):
"""
Wraps Singular's invariant_algebra_reynolds and invariant_ring
in finvar.lib, with help from Simon King and Martin Albrecht.
Computes generators for the polynomial ring
`F[x_1,\ldots,x_n]^G`, where G in GL(n,F) is a finite
matrix group.
In the "good characteristic" case the polynomials returned form a
minimal generating set for the algebra of G-invariant polynomials.
In the "bad" case, the polynomials returned are primary and
secondary invariants, forming a not necessarily minimal generating
set for the algebra of G-invariant polynomials.
EXAMPLES::
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()
[x1^7*x2 - x1*x2^7, x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12, x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; gen2 = [[1,0],[0,-1]]; gen3 = [[-1,0],[0,1]]
sage: G = MatrixGroup([MS(gen1),MS(gen2),MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,2],[-1,1]]),MS([[1,1],[-1,1]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators() # long time (64s on sage.math, 2011)
[x1^20 + x1^16*x2^4 + x1^12*x2^8 + x1^8*x2^12 + x1^4*x2^16 + x2^20, x1^20*x2^4 + x1^16*x2^8 + x1^12*x2^12 + x1^8*x2^16 + x1^4*x2^20]
sage: F=CyclotomicField(8)
sage: z=F.gen()
sage: a=z+1/z
sage: b=z^2
sage: MS=MatrixSpace(F,2,2)
sage: g1=MS([[1/a,1/a],[1/a,-1/a]])
sage: g2=MS([[1,0],[0,b]])
sage: g3=MS([[b,0],[0,1]])
sage: G=MatrixGroup([g1,g2,g3])
sage: G.invariant_generators() # long time (12s on sage.math, 2011)
[x1^8 + 14*x1^4*x2^4 + x2^8,
x1^24 + 10626/1025*x1^20*x2^4 + 735471/1025*x1^16*x2^8 + 2704156/1025*x1^12*x2^12 + 735471/1025*x1^8*x2^16 + 10626/1025*x1^4*x2^20 + x2^24]
AUTHORS:
- David Joyner, Simon King and Martin Albrecht.
REFERENCES:
- Singular reference manual
- B. Sturmfels, "Algorithms in invariant theory", Springer-Verlag, 1993.
- S. King, "Minimal Generating Sets of non-modular invariant rings of
finite groups", arXiv:math.AC/0703035
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.interfaces.singular import singular
gens = self.gens()
singular.LIB("finvar.lib")
n = self.degree() #len((gens[0].matrix()).rows())
F = self.base_ring()
q = F.characteristic()
## test if the field is admissible
if F.gen()==1: # we got the rationals or GF(prime)
FieldStr = str(F.characteristic())
elif hasattr(F,'polynomial'): # we got an algebraic extension
if len(F.gens())>1:
raise NotImplementedError, "can only deal with finite fields and (simple algebraic extensions of) the rationals"
FieldStr = '(%d,%s)'%(F.characteristic(),str(F.gen()))
else: # we have a transcendental extension
FieldStr = '(%d,%s)'%(F.characteristic(),','.join([str(p) for p in F.gens()]))
## Setting Singular's variable names
## We need to make sure that field generator and variables get different names.
if str(F.gen())[0]=='x':
VarStr = 'y'
else:
VarStr = 'x'
VarNames='('+','.join((VarStr+str(i+1) for i in range(n)))+')'
R=singular.ring(FieldStr,VarNames,'dp')
if hasattr(F,'polynomial') and F.gen()!=1: # we have to define minpoly
singular.eval('minpoly = '+str(F.polynomial()).replace('x',str(F.gen())))
A = [singular.matrix(n,n,str((x.matrix()).list())) for x in gens]
Lgens = ','.join((x.name() for x in A))
PR = PolynomialRing(F,n,[VarStr+str(i) for i in range(1,n+1)])
if q == 0 or (q > 0 and self.cardinality()%q != 0):
from sage.all import Integer, Matrix
try:
gapself = gap(self)
# test whether the backwards transformation works as well:
for i in range(self.ngens()):
if Matrix(gapself.gen(i+1),F) != self.gen(i).matrix():
raise ValueError
except (TypeError,ValueError):
gapself is None
if gapself is not None:
ReyName = 't'+singular._next_var_name()
singular.eval('matrix %s[%d][%d]'%(ReyName,self.cardinality(),n))
En = gapself.Enumerator()
for i in range(1,self.cardinality()+1):
M = Matrix(En[i],F)
D = [{} for foobar in range(self.degree())]
for x,y in M.dict().items():
D[x[0]][x[1]] = y
for row in range(self.degree()):
for t in D[row].items():
singular.eval('%s[%d,%d]=%s[%d,%d]+(%s)*var(%d)'%(ReyName,i,row+1,ReyName,i,row+1, repr(t[1]),t[0]+1))
foobar = singular(ReyName)
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s)'%(IRName,ReyName))
else:
ReyName = 't'+singular._next_var_name()
singular.eval('list %s=group_reynolds((%s))'%(ReyName,Lgens))
IRName = 't'+singular._next_var_name()
singular.eval('matrix %s = invariant_algebra_reynolds(%s[1])'%(IRName,ReyName))
OUT = [singular.eval(IRName+'[1,%d]'%(j)) for j in range(1,1+singular('ncols('+IRName+')'))]
return [PR(gen) for gen in OUT]
if self.cardinality()%q == 0:
PName = 't'+singular._next_var_name()
SName = 't'+singular._next_var_name()
singular.eval('matrix %s,%s=invariant_ring(%s)'%(PName,SName,Lgens))
OUT = [singular.eval(PName+'[1,%d]'%(j)) for j in range(1,1+singular('ncols('+PName+')'))] + [singular.eval(SName+'[1,%d]'%(j)) for j in range(2,1+singular('ncols('+SName+')'))]
return [PR(gen) for gen in OUT]
