def _singular_(self, singular=singular):
r"""
Returns a singular ring for this polynomial ring.
Currently `\QQ`, `{\rm GF}(p), {\rm GF}(p^n)`, `\CC`, `\RR`, `\ZZ` and
`\ZZ/n\ZZ` are supported.
INPUT:
- ``singular`` - Singular instance
OUTPUT: Singular ring matching this ring
EXAMPLES::
sage: R.<x,y> = PolynomialRing(CC,'x',2)
sage: singular(R)
// characteristic : 0 (complex:15 digits, additional 0 digits)
// 1 parameter : I
// minpoly : (I^2+1)
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R.<x,y> = PolynomialRing(RealField(100),'x',2)
sage: singular(R)
// characteristic : 0 (real:29 digits, additional 0 digits)
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: w = var('w')
sage: R.<x> = PolynomialRing(NumberField(w^2+1,'s'))
sage: singular(R)
// characteristic : 0
// 1 parameter : s
// minpoly : (s^2+1)
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(GF(127),1,'x')
sage: singular(R)
// characteristic : 127
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(QQ,1,'x')
sage: singular(R)
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(QQ,'x')
sage: singular(R)
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(GF(127),'x')
sage: singular(R)
// characteristic : 127
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = Frac(ZZ['a,b'])['x,y']
sage: singular(R)
// characteristic : 0
// 2 parameter : a b
// minpoly : 0
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = IntegerModRing(1024)['x,y']
sage: singular(R)
// coeff. ring is : Z/2^10
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = IntegerModRing(15)['x,y']
sage: singular(R)
// coeff. ring is : Z/15
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = ZZ['x,y']
sage: singular(R)
// coeff. ring is : Integers
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: k.<a> = FiniteField(25)
sage: R = k['x']
sage: K = R.fraction_field()
sage: S = K['y']
sage: singular(S)
// characteristic : 5
// 1 parameter : x
// minpoly : 0
// number of vars : 2
// block 1 : ordering lp
// : names a y
// block 2 : ordering C
// quotient ring from ideal
_[1]=a2-a+2
.. warning::
- If the base ring is a finite extension field or a number field
the ring will not only be returned but also be set as the current
ring in Singular.
- Singular represents precision of floating point numbers base 10
while Sage represents floating point precision base 2.
"""
try:
R = self.__singular
if not (R.parent() is singular):
raise ValueError
R._check_valid()
if self.base_ring() is ZZ or self.base_ring().is_prime_field():
return R
if sage.rings.finite_rings.finite_field_constructor.is_FiniteField(self.base_ring()) or\
(number_field.number_field_base.is_NumberField(self.base_ring()) and self.base_ring().is_absolute()):
R.set_ring() #sorry for that, but needed for minpoly
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly))
self.__minpoly = singular.eval('minpoly')[1:-1]
return R
except (AttributeError, ValueError):
return self._singular_init_(singular)
def _singular_(self, singular=singular):
r"""
Returns a singular ring for this polynomial ring.
Currently `\QQ`, `{\rm GF}(p), {\rm GF}(p^n)`, `\CC`, `\RR`, `\ZZ` and
`\ZZ/n\ZZ` are supported.
INPUT:
- ``singular`` - Singular instance
OUTPUT: Singular ring matching this ring
EXAMPLES::
sage: R.<x,y> = PolynomialRing(CC,'x',2)
sage: singular(R)
// characteristic : 0 (complex:15 digits, additional 0 digits)
// 1 parameter : I
// minpoly : (I^2+1)
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R.<x,y> = PolynomialRing(RealField(100),'x',2)
sage: singular(R)
// characteristic : 0 (real:29 digits, additional 0 digits)
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: w = var('w')
sage: R.<x> = PolynomialRing(NumberField(w^2+1,'s'))
sage: singular(R)
// characteristic : 0
// 1 parameter : s
// minpoly : (s^2+1)
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(GF(127),1,'x')
sage: singular(R)
// characteristic : 127
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(QQ,1,'x')
sage: singular(R)
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(QQ,'x')
sage: singular(R)
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = PolynomialRing(GF(127),'x')
sage: singular(R)
// characteristic : 127
// number of vars : 1
// block 1 : ordering lp
// : names x
// block 2 : ordering C
sage: R = Frac(ZZ['a,b'])['x,y']
sage: singular(R)
// characteristic : 0
// 2 parameter : a b
// minpoly : 0
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = IntegerModRing(1024)['x,y']
sage: singular(R)
// coeff. ring is : Z/2^10
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = IntegerModRing(15)['x,y']
sage: singular(R)
// coeff. ring is : Z/15
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: R = ZZ['x,y']
sage: singular(R)
// coeff. ring is : Integers
// number of vars : 2
// block 1 : ordering dp
// : names x y
// block 2 : ordering C
sage: k.<a> = FiniteField(25)
sage: R = k['x']
sage: K = R.fraction_field()
sage: S = K['y']
sage: singular(S)
// characteristic : 5
// 1 parameter : x
// minpoly : 0
// number of vars : 2
// block 1 : ordering lp
// : names a y
// block 2 : ordering C
// quotient ring from ideal
_[1]=a2-a+2
.. warning::
- If the base ring is a finite extension field or a number field
the ring will not only be returned but also be set as the current
ring in Singular.
- Singular represents precision of floating point numbers base 10
while Sage represents floating point precision base 2.
"""
try:
R = self.__singular
if not (R.parent() is singular):
raise ValueError
R._check_valid()
if self.base_ring() is ZZ or self.base_ring().is_prime_field():
return R
if sage.rings.finite_rings.constructor.is_FiniteField(self.base_ring()) or\
(number_field.number_field_base.is_NumberField(self.base_ring()) and self.base_ring().is_absolute()):
R.set_ring() #sorry for that, but needed for minpoly
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly))
self.__minpoly = singular.eval('minpoly')[1:-1]
return R
except (AttributeError, ValueError):
return self._singular_init_(singular)
def _singular_init_(self, singular=singular):
"""
Return a newly created Singular ring matching this ring.
EXAMPLES::
sage: PolynomialRing(QQ,'u_ba')._singular_init_()
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names u_ba
// block 2 : ordering C
"""
if not can_convert_to_singular(self):
raise TypeError("no conversion of this ring to a Singular ring defined")
if self.ngens()==1:
_vars = '(%s)'%self.gen()
if "*" in _vars: # 1.000...000*x
_vars = _vars.split("*")[1]
order = 'lp'
else:
_vars = str(self.gens())
order = self.term_order().singular_str()
base_ring = self.base_ring()
if is_RealField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.arith.all.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order, check=False)
elif is_ComplexField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.arith.all.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order, check=False)
elif is_RealDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(real,15,0)", _vars, order=order, check=False)
elif is_ComplexDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(complex,15,0,I)", _vars, order=order, check=False)
elif base_ring.is_prime_field():
self.__singular = singular.ring(self.characteristic(), _vars, order=order, check=False)
elif sage.rings.finite_rings.finite_field_constructor.is_FiniteField(base_ring):
# not the prime field!
gen = str(base_ring.gen())
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (str(base_ring.modulus()).replace("x",gen)).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif number_field.number_field_base.is_NumberField(base_ring) and base_ring.is_absolute():
# not the rationals!
gen = str(base_ring.gen())
poly=base_ring.polynomial()
poly_gen=str(poly.parent().gen())
poly_str=str(poly).replace(poly_gen,gen)
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (poly_str).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif sage.rings.fraction_field.is_FractionField(base_ring) and (base_ring.base_ring() is ZZ or base_ring.base_ring().is_prime_field() or is_FiniteField(base_ring.base_ring())):
if base_ring.ngens()==1:
gens = str(base_ring.gen())
else:
gens = str(base_ring.gens())
if not (not base_ring.base_ring().is_prime_field() and is_FiniteField(base_ring.base_ring())) :
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
else:
ext_gen = str(base_ring.base_ring().gen())
_vars = '(' + ext_gen + ', ' + _vars[1:];
R = self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
self.base_ring().__minpoly = (str(base_ring.base_ring().modulus()).replace("x",ext_gen)).replace(" ","")
singular.eval('setring '+R._name);
self.__singular = singular("std(ideal(%s))"%(self.base_ring().__minpoly),type='qring')
elif sage.rings.function_field.function_field.is_RationalFunctionField(base_ring) and base_ring.constant_field().is_prime_field():
gen = str(base_ring.gen())
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gen), _vars, order=order, check=False)
elif is_IntegerModRing(base_ring):
ch = base_ring.characteristic()
if ch.is_power_of(2):
exp = ch.nbits() -1
self.__singular = singular.ring("(integer,2,%d)"%(exp,), _vars, order=order, check=False)
else:
self.__singular = singular.ring("(integer,%d)"%(ch,), _vars, order=order, check=False)
elif base_ring is ZZ:
self.__singular = singular.ring("(integer)", _vars, order=order, check=False)
else:
raise TypeError("no conversion to a Singular ring defined")
return self.__singular
def _singular_init_(self, singular=singular):
"""
Return a newly created Singular ring matching this ring.
EXAMPLES::
sage: PolynomialRing(QQ,'u_ba')._singular_init_()
// characteristic : 0
// number of vars : 1
// block 1 : ordering lp
// : names u_ba
// block 2 : ordering C
"""
if not can_convert_to_singular(self):
raise TypeError("no conversion of this ring to a Singular ring defined")
if self.ngens()==1:
_vars = '(%s)'%self.gen()
if "*" in _vars: # 1.000...000*x
_vars = _vars.split("*")[1]
order = 'lp'
else:
_vars = str(self.gens())
order = self.term_order().singular_str()
base_ring = self.base_ring()
if is_RealField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(real,%d,0)"%digits, _vars, order=order, check=False)
elif is_ComplexField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
precision = base_ring.precision()
digits = sage.rings.arith.integer_ceil((2*precision - 2)/7.0)
self.__singular = singular.ring("(complex,%d,0,I)"%digits, _vars, order=order, check=False)
elif is_RealDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(real,15,0)", _vars, order=order, check=False)
elif is_ComplexDoubleField(base_ring):
# singular converts to bits from base_10 in mpr_complex.cc by:
# size_t bits = 1 + (size_t) ((float)digits * 3.5);
self.__singular = singular.ring("(complex,15,0,I)", _vars, order=order, check=False)
elif base_ring.is_prime_field():
self.__singular = singular.ring(self.characteristic(), _vars, order=order, check=False)
elif sage.rings.finite_rings.constructor.is_FiniteField(base_ring):
# not the prime field!
gen = str(base_ring.gen())
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (str(base_ring.modulus()).replace("x",gen)).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif number_field.number_field_base.is_NumberField(base_ring) and base_ring.is_absolute():
# not the rationals!
gen = str(base_ring.gen())
poly=base_ring.polynomial()
poly_gen=str(poly.parent().gen())
poly_str=str(poly).replace(poly_gen,gen)
r = singular.ring( "(%s,%s)"%(self.characteristic(),gen), _vars, order=order, check=False)
self.__minpoly = (poly_str).replace(" ","")
if singular.eval('minpoly') != "(" + self.__minpoly + ")":
singular.eval("minpoly=%s"%(self.__minpoly) )
self.__minpoly = singular.eval('minpoly')[1:-1]
self.__singular = r
elif sage.rings.fraction_field.is_FractionField(base_ring) and (base_ring.base_ring() is ZZ or base_ring.base_ring().is_prime_field() or is_FiniteField(base_ring.base_ring())):
if base_ring.ngens()==1:
gens = str(base_ring.gen())
else:
gens = str(base_ring.gens())
if not (not base_ring.base_ring().is_prime_field() and is_FiniteField(base_ring.base_ring())) :
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
else:
ext_gen = str(base_ring.base_ring().gen())
_vars = '(' + ext_gen + ', ' + _vars[1:];
R = self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gens), _vars, order=order, check=False)
self.base_ring().__minpoly = (str(base_ring.base_ring().modulus()).replace("x",ext_gen)).replace(" ","")
singular.eval('setring '+R._name);
self.__singular = singular("std(ideal(%s))"%(self.base_ring().__minpoly),type='qring')
elif sage.rings.function_field.function_field.is_RationalFunctionField(base_ring) and base_ring.constant_field().is_prime_field():
gen = str(base_ring.gen())
self.__singular = singular.ring( "(%s,%s)"%(base_ring.characteristic(),gen), _vars, order=order, check=False)
elif is_IntegerModRing(base_ring):
ch = base_ring.characteristic()
if ch.is_power_of(2):
exp = ch.nbits() -1
self.__singular = singular.ring("(integer,2,%d)"%(exp,), _vars, order=order, check=False)
else:
self.__singular = singular.ring("(integer,%d)"%(ch,), _vars, order=order, check=False)
elif base_ring is ZZ:
self.__singular = singular.ring("(integer)", _vars, order=order, check=False)
else:
raise TypeError("no conversion to a Singular ring defined")
return self.__singular
