Python PolynomialRing.coerce方法代码示例

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import coerce [as 别名]
    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

# 需要导入模块: from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing [as 别名]
# 或者: from sage.rings.polynomial.polynomial_ring_constructor.PolynomialRing import coerce [as 别名]
    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:
#.........这里部分代码省略.........