def is_morphism(self):
r"""
returns ``True`` if self is a morphism (no common zero of defining polynomials).
The map is a morphism if and only if the ideal generated by the defining polynomials
is the unit ideal.
OUTPUT:
- Boolean
EXAMPLES::
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([x^2+y^2,y^2])
sage: f.is_morphism()
True
::
sage: P.<x,y,z>=ProjectiveSpace(RR,2)
sage: H=Hom(P,P)
sage: f=H([x*z-y*z,x^2-y^2,z^2])
sage: f.is_morphism()
False
::
sage: R.<t>=PolynomialRing(GF(5))
sage: P.<x,y,z>=ProjectiveSpace(R,2)
sage: H=Hom(P,P)
sage: f=H([x*z-t*y^2,x^2-y^2,t*z^2])
sage: f.is_morphism()
True
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if is_ProjectiveSpace(self.domain())==False or is_ProjectiveSpace(self.codomain())==False:
raise NotImplementedError
R=self.coordinate_ring()
F=self._polys
if R.base_ring().is_field():
J=R.ideal(F)
else:
S=PolynomialRing(R.base_ring().fraction_field(),R.gens(),R.ngens())
J=S.ideal([S.coerce(F[i]) for i in range(R.ngens())])
if J.dimension()>0:
return False
else:
return True
def primes_of_bad_reduction(self, check=True):
r"""
Determines the primes of bad reduction for a map `self: \mathbb{P}^N \to \mathbb{P}^N`
defined over `\ZZ` or `\QQ`.
If ``check`` is ``True``, each prime is verified to be of bad reduction.
ALGORITHM:
`p` is a prime of bad reduction if and only if the defining polynomials of self
have a common zero. Or stated another way, `p` is a prime of bad reducion if and only if the
radical of the ideal defined by the defining polynomials of self is not `(x_0,x_1,\ldots,x_N)`.
This happens if and only if some power of each `x_i` is not in the ideal defined by the defining
polynomials of self. This last condition is what is checked. The lcm of the coefficients of
the monomials `x_i` in a groebner basis is computed. This may return extra primes.
INPUT:
- ``check`` -- Boolean (optional - default: ``True``)
OUTPUT:
- a list of integer primes.
EXAMPLES::
sage: P.<x,y>=ProjectiveSpace(QQ,1)
sage: H=Hom(P,P)
sage: f=H([1/3*x^2+1/2*y^2,y^2])
sage: print f.primes_of_bad_reduction()
[2, 3]
::
sage: P.<x,y,z,w>=ProjectiveSpace(QQ,3)
sage: H=Hom(P,P)
sage: f=H([12*x*z-7*y^2,31*x^2-y^2,26*z^2,3*w^2-z*w])
sage: f.primes_of_bad_reduction()
[2, 3, 7, 13, 31]
::
This is an example where check=False returns extra primes
sage: P.<x,y,z>=ProjectiveSpace(ZZ,2)
sage: H=Hom(P,P)
sage: f=H([3*x*y^2 + 7*y^3 - 4*y^2*z + 5*z^3, -5*x^3 + x^2*y + y^3 + 2*x^2*z, -2*x^2*y + x*y^2 + y^3 - 4*y^2*z + x*z^2])
sage: f.primes_of_bad_reduction(False)
[2, 5, 37, 2239, 304432717]
sage: f.primes_of_bad_reduction()
[5, 37, 2239, 304432717]
"""
if self.base_ring() != ZZ and self.base_ring() != QQ:
raise TypeError("Must be ZZ or QQ")
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if is_ProjectiveSpace(self.domain())==False or is_ProjectiveSpace(self.codomain())==False:
raise NotImplementedError
R=self.coordinate_ring()
F=self._polys
if R.base_ring().is_field():
J=R.ideal(F)
else:
S=PolynomialRing(R.base_ring().fraction_field(),R.gens(),R.ngens())
J=S.ideal([S.coerce(F[i]) for i in range(R.ngens())])
if J.dimension()>0:
raise TypeError("Not a morphism.")
#normalize to coefficients in the ring not the fraction field.
F=[F[i]*lcm([F[j].denominator() for j in range(len(F))]) for i in range(len(F))]
#move the ideal to the ring of integers
if R.base_ring().is_field():
S=PolynomialRing(R.base_ring().ring_of_integers(),R.gens(),R.ngens())
F=[F[i].change_ring(R.base_ring().ring_of_integers()) for i in range(len(F))]
J=S.ideal(F)
else:
J=R.ideal(F)
GB=J.groebner_basis()
badprimes=[]
#get the primes dividing the coefficients of the monomials x_i^k_i
for i in range(len(GB)):
LT=GB[i].lt().degrees()
power=0
for j in range(R.ngens()):
if LT[j]!=0:
power+=1
if power==1:
badprimes=badprimes+GB[i].lt().coefficients()[0].support()
badprimes=list(set(badprimes))
badprimes.sort()
#check to return only the truly bad primes
if check==True:
index=0
while index < len(badprimes): #figure out which primes are really bad primes...
S=PolynomialRing(GF(badprimes[index]),R.gens(),R.ngens())
J=S.ideal([S.coerce(F[j]) for j in range(R.ngens())])
if J.dimension()==0:
badprimes.pop(index)
else:
index+=1
return(badprimes)
