affine_point.py

r"""
Points on affine varieties

This module implements scheme morphism for points on affine varieties.

AUTHORS:

- David Kohel, William Stein (2006): initial version
- Volker Braun (2011-08-08): renamed classes, more documentation, misc cleanups
- Ben Hutz (2013): many improvements
"""

# ****************************************************************************
#       Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu.au>
#       Copyright (C) 2006 William Stein <wstein@gmail.com>
#       Copyright (C) 2011 Volker Braun <vbraun.name@gmail.com>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#  the License, or (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

from sage.categories.number_fields import NumberFields
from sage.misc.lazy_import import lazy_import
from sage.rings.integer_ring import ZZ
from sage.schemes.generic.morphism import SchemeMorphism_point, SchemeMorphism
from sage.structure.sequence import Sequence

_NumberFields = NumberFields()


# --------------------------------------------------------------------
# Rational points on schemes, which we view as morphisms determined by
# coordinates.
# --------------------------------------------------------------------

class SchemeMorphism_point_affine(SchemeMorphism_point):
    """
    A rational point on an affine scheme.

    INPUT:

    - ``X`` -- a subscheme of an ambient affine space over a ring `R`

    - ``v`` -- list/tuple/iterable of coordinates in `R`

    - ``check`` -- boolean (default: ``True``); whether to
      check the input for consistency

    EXAMPLES::

        sage: A = AffineSpace(2, QQ)
        sage: A(1, 2)
        (1, 2)
    """
    def __init__(self, X, v, check=True):
        """
        The Python constructor.

        See :class:`SchemeMorphism_point_affine` for details.

        TESTS::

            sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
            sage: A3.<x,y,z> = AffineSpace(QQ, 3)
            sage: SchemeMorphism_point_affine(A3(QQ), [1, 2, 3])
            (1, 2, 3)
        """
        SchemeMorphism.__init__(self, X)
        if check:
            from sage.categories.commutative_rings import CommutativeRings
            if isinstance(v, SchemeMorphism):
                v = list(v)
            else:
                try:
                    if v.parent() in CommutativeRings():
                        v = [v]
                except AttributeError:
                    pass
            # Verify that there are the right number of coords
            d = self.codomain().ambient_space().ngens()
            if len(v) != d:
                raise TypeError("argument v (=%s) must have %s coordinates" % (v, d))
            if not isinstance(v, (list, tuple)):
                raise TypeError("argument v (= %s) must be a scheme point, list, or tuple" % str(v))
            # Make sure the coordinates all lie in the appropriate ring
            v = Sequence(v, X.value_ring())
            # Verify that the point satisfies the equations of X.
            X.extended_codomain()._check_satisfies_equations(v)
        self._coords = tuple(v)

    def _matrix_times_point_(self, mat, dom):
        r"""
        Multiply the point on the left by a matrix ``mat``.

        INPUT:

        - ``mat`` -- a matrix

        - ``dom`` -- (unused) needed for consistent function call with projective

        OUTPUT: a scheme point given by ``mat*self``

        EXAMPLES::

            sage: P = AffineSpace(QQ,2)
            sage: Q = P(1,2)
            sage: m = matrix(ZZ, 3, 3, [0,1,1,0,0,1,1,1,1])                             # needs sage.modules
            sage: m*Q                                                                   # needs sage.modules
            (3/4, 1/4)

        ::

            sage: P = AffineSpace(QQ,1)
            sage: Q = P(0)
            sage: m = matrix(RR, 2, 2, [0,1,1,0])                                       # needs sage.modules
            sage: m*Q                                                                   # needs sage.modules
            Traceback (most recent call last):
            ...
            ValueError: resulting point not affine

        ::

            sage: P = AffineSpace(QQ,2)
            sage: Q = P(1,1)
            sage: m = matrix(RR, 2, 2, [0,1,1,0])                                       # needs sage.modules
            sage: m*Q                                                                   # needs sage.modules
            Traceback (most recent call last):
            ...
            ValueError: matrix size is incompatible
        """
        # input checking done in projective implementation
        d = self.codomain().ngens()
        P = mat * self.homogenize(d)
        if P[-1] == 0:
            raise ValueError("resulting point not affine")
        return P.dehomogenize(d)

    def __hash__(self):
        r"""
        Compute the hash value of this affine point.

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: hash(A([1, 1])) == hash((1,1))
            True

        ::

            sage: A.<x,y,z> = AffineSpace(CC, 3)                                        # needs sage.rings.real_mpfr
            sage: pt = A([1, 2, -i])                                                    # needs sage.rings.real_mpfr sage.symbolic
            sage: hash(pt) == hash(tuple(pt))                                           # needs sage.rings.real_mpfr sage.symbolic
            True
        """
        return hash(tuple(self))

    def global_height(self, prec=None):
        r"""
        Return the logarithmic height of the point.

        INPUT:

        - ``prec`` -- desired floating point precision (default:
          default RealField precision)

        OUTPUT: a real number

        EXAMPLES::

            sage: P.<x,y> = AffineSpace(QQ, 2)
            sage: Q = P(41, 1/12)
            sage: Q.global_height()                                                     # needs sage.rings.real_mpfr
            3.71357206670431

        ::

            sage: P = AffineSpace(ZZ, 4, 'x')
            sage: Q = P(3, 17, -51, 5)
            sage: Q.global_height()                                                     # needs sage.rings.real_mpfr
            3.93182563272433

        ::

            sage: R.<x> = PolynomialRing(QQ)
            sage: k.<w> = NumberField(x^2 + 5)                                          # needs sage.rings.number_field
            sage: A = AffineSpace(k, 2, 'z')                                            # needs sage.rings.number_field
            sage: A([3, 5*w + 1]).global_height(prec=100)                               # needs sage.rings.number_field sage.rings.real_mpfr
            2.4181409534757389986565376694

        .. TODO::

            P-adic heights.
        """
        if self.domain().base_ring() == ZZ:
            from sage.rings.real_mpfr import RealField
            if prec is None:
                R = RealField()
            else:
                R = RealField(prec)
            H = max([self[i].abs() for i in range(self.codomain().ambient_space().dimension_relative())])
            return R(max(H, 1)).log()
        if self.domain().base_ring() in _NumberFields or isinstance(self.domain().base_ring(), sage.rings.abc.Order):
            return max([self[i].global_height(prec) for i in range(self.codomain().ambient_space().dimension_relative())])
        else:
            raise NotImplementedError("must be over a number field or a number field Order")

    def homogenize(self, n):
        r"""
        Return the homogenization of the point at the ``nth`` coordinate.

        INPUT:

        - ``n`` -- integer between 0 and dimension of the map, inclusive

        OUTPUT: a point in the projectivization of the codomain of the map

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(ZZ, 2)
            sage: Q = A(2, 3)
            sage: Q.homogenize(2).dehomogenize(2) == Q
            True

            ::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: Q = A(2, 3)
            sage: P = A(0, 1)
            sage: Q.homogenize(2).codomain() == P.homogenize(2).codomain()
            True
        """
        phi = self.codomain().projective_embedding(n)
        return phi(self)


class SchemeMorphism_point_affine_field(SchemeMorphism_point_affine):

    def __hash__(self):
        r"""
        Compute the hash value of this affine point.

        EXAMPLES::

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: X = A.subscheme(x - y)
            sage: hash(X([1, 1])) == hash((1,1))
            True

            sage: A.<x,y> = AffineSpace(QQ, 2)
            sage: X = A.subscheme(x^2 - y^3)
            sage: pt = X([1, 1])
            sage: hash(pt) == hash(tuple(pt))
            True
        """
        return hash(tuple(self))

    def weil_restriction(self):
        r"""
        Compute the Weil restriction of this point over some extension
        field.

        If the field is a finite field, then this computes
        the Weil restriction to the prime subfield.

        A Weil restriction of scalars - denoted `Res_{L/k}` - is a
        functor which, for any finite extension of fields `L/k` and
        any algebraic variety `X` over `L`, produces another
        corresponding variety `Res_{L/k}(X)`, defined over `k`. It is
        useful for reducing questions about varieties over large
        fields to questions about more complicated varieties over
        smaller fields. This functor applied to a point gives
        the equivalent point on the Weil restriction of its
        codomain.

        OUTPUT: scheme point on the Weil restriction of the codomain of this point

        EXAMPLES::

            sage: # needs sage.libs.singular sage.rings.finite_rings
            sage: A.<x,y,z> = AffineSpace(GF(5^3, 't'), 3)
            sage: X = A.subscheme([y^2 - x*z, z^2 + y])
            sage: Y = X.weil_restriction()
            sage: P = X([1, -1, 1])
            sage: Q = P.weil_restriction();Q
            (1, 0, 0, 4, 0, 0, 1, 0, 0)
            sage: Q.codomain() == Y
            True

        ::

            sage: # needs sage.libs.singular sage.rings.number_field
            sage: R.<x> = QQ[]
            sage: K.<w> = NumberField(x^5 - 2)
            sage: R.<x> = K[]
            sage: L.<v> = K.extension(x^2 + w)
            sage: A.<x,y> = AffineSpace(L, 2)
            sage: P = A([w^3 - v, 1 + w + w*v])
            sage: P.weil_restriction()
            (w^3, -1, w + 1, w)
        """
        L = self.codomain().base_ring()
        WR = self.codomain().weil_restriction()
        if L.is_finite():
            d = L.degree()
            if d == 1:
                return self
            newP = []
            for t in self:
                c = t.polynomial().coefficients(sparse=False)
                c = c + (d - len(c)) * [0]
                newP += c
        else:
            d = L.relative_degree()
            if d == 1:
                return self
            # create a CoordinateFunction that gets the relative coordinates in terms of powers
            from sage.rings.number_field.number_field_element import CoordinateFunction
            v = L.gen()
            V, from_V, to_V = L.relative_vector_space()
            h = L(1)
            B = [to_V(h)]
            f = v.minpoly()
            for i in range(f.degree() - 1):
                h *= v
                B.append(to_V(h))
            W = V.span_of_basis(B)
            p = CoordinateFunction(v, W, to_V)
            newP = []
            for t in self:
                newP += p(t)
        return WR(newP)

    def intersection_multiplicity(self, X):
        r"""
        Return the intersection multiplicity of the codomain of this point and ``X`` at this point.

        This uses the intersection_multiplicity implementations for projective/affine subschemes. This
        point must be a point on an affine subscheme.

        INPUT:

        - ``X`` -- a subscheme in the same ambient space as that of the codomain of this point

        OUTPUT: integer

        EXAMPLES::

            sage: # needs sage.libs.singular
            sage: A.<x,y> = AffineSpace(GF(17), 2)
            sage: X = A.subscheme([y^2 - x^3 + 2*x^2 - x])
            sage: Y = A.subscheme([y - 2*x + 2])
            sage: Q1 = Y([1,0])
            sage: Q1.intersection_multiplicity(X)
            2
            sage: Q2 = X([4,6])
            sage: Q2.intersection_multiplicity(Y)
            1

        ::

            sage: A.<x,y,z,w> = AffineSpace(QQ, 4)
            sage: X = A.subscheme([x^2 - y*z^2, z - 2*w^2])
            sage: Q = A([2,1,2,-1])
            sage: Q.intersection_multiplicity(X)
            Traceback (most recent call last):
            ...
            TypeError: this point must be a point on an affine subscheme
        """
        from sage.schemes.affine.affine_space import AffineSpace_generic
        if isinstance(self.codomain(), AffineSpace_generic):
            raise TypeError("this point must be a point on an affine subscheme")
        return self.codomain().intersection_multiplicity(X, self)

    def multiplicity(self):
        r"""
        Return the multiplicity of this point on its codomain.

        Uses the subscheme multiplicity implementation. This point must be a point on an
        affine subscheme.

        OUTPUT: integer

        EXAMPLES::

            sage: A.<x,y,z> = AffineSpace(QQ, 3)
            sage: X = A.subscheme([y^2 - x^7*z])
            sage: Q1 = X([1,1,1])
            sage: Q1.multiplicity()                                                     # needs sage.libs.singular
            1
            sage: Q2 = X([0,0,2])
            sage: Q2.multiplicity()                                                     # needs sage.libs.singular
            2
        """
        from sage.schemes.affine.affine_space import AffineSpace_generic
        if isinstance(self.codomain(), AffineSpace_generic):
            raise TypeError("this point must be a point on an affine subscheme")
        return self.codomain().multiplicity(self)

    def as_subscheme(self):
        r"""
        Return the subscheme associated with this rational point.

        EXAMPLES::

            sage: A2.<x,y> = AffineSpace(QQ, 2)
            sage: p1 = A2.point([0,0]).as_subscheme(); p1
            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
              x, y
            sage: p2 = A2.point([1,1]).as_subscheme(); p2
            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
              x - 1, y - 1
            sage: p1 + p2
            Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
              x - y, y^2 - y
        """
        A = self.codomain().ambient_space()
        g = A.gens()
        v = self._coords
        n = len(v)
        return A.subscheme([g[i] - v[i] for i in range(n)])


class SchemeMorphism_point_affine_finite_field(SchemeMorphism_point_affine_field):

    def __hash__(self):
        r"""
        Return the integer hash of the point.

        OUTPUT: integer

        EXAMPLES::

            sage: P.<x,y,z> = AffineSpace(GF(5), 3)
            sage: hash(P(2, 1, 2))
            57

        ::

            sage: P.<x,y,z> = AffineSpace(GF(7), 3)
            sage: X = P.subscheme(x^2 - y^2)
            sage: hash(X(1, 1, 2))
            106

        ::

            sage: P.<x,y> = AffineSpace(GF(13), 2)
            sage: hash(P(3, 4))
            55

        ::

            sage: P.<x,y> = AffineSpace(GF(13^3, 't'), 2)                               # needs sage.rings.finite_rings
            sage: hash(P(3, 4))                                                         # needs sage.rings.finite_rings
            8791
        """
        p = self.codomain().base_ring().order()
        N = self.codomain().ambient_space().dimension_relative()
        return int(sum(hash(self[i]) * p**i for i in range(N)))