예제 #1
0
def complementary_subform_to_vector(self, v):
    """
    Finds the `(n-1)`-dim'l quadratic form orthogonal to the vector `v`.

    Note: This is usually not a direct summand!

    Technical Notes: There is a minor difference in the cancellation
    code here (form the C++ version) since the notation Q `[i,j]` indexes
    coefficients of the quadratic polynomial here, not the symmetric
    matrix.  Also, it produces a better splitting now, for the full
    lattice (as opposed to a sublattice in the C++ code) since we
    now extend `v` to a unimodular matrix.

    INPUT:

        `v` -- a list of self.dim() integers

    OUTPUT:

        a QuadraticForm over `ZZ`


    EXAMPLES::

        sage: Q1 = DiagonalQuadraticForm(ZZ, [1,3,5,7])
        sage: Q1.complementary_subform_to_vector([1,0,0,0])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 3 0 0 ]
        [ * 5 0 ]
        [ * * 7 ]

    ::

        sage: Q1.complementary_subform_to_vector([1,1,0,0])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 12 0 0 ]
        [ * 5 0 ]
        [ * * 7 ]

    ::

        sage: Q1.complementary_subform_to_vector([1,1,1,1])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 624 -480 -672 ]
        [ * 880 -1120 ]
        [ * * 1008 ]

    """
    n = self.dim()

    ## Copy the quadratic form
    Q = deepcopy(self)

    ## Find the first non-zero component of v, and call it nz  (Note: 0 <= nz < n)
    nz = 0
    while (nz < n) and (v[nz] == 0):
        nz += 1

    ## Abort if v is the zero vector
    if nz == n:
        raise TypeError("Oops, v cannot be the zero vector! =(")

    ## Make the change of basis matrix
    new_basis = extend_to_primitive(matrix(ZZ, n, 1, v))

    ## Change Q (to Q1) to have v as its nz-th basis vector
    Q1 = Q(new_basis)

    ## Pick out the value Q(v) of the vector
    d = Q1[0, 0]

    ## For each row/column, perform elementary operations to cancel them out.
    for i in range(1, n):

        ## Check if the (i,0)-entry is divisible by d,
        ## and stretch its row/column if not.
        if Q1[i, 0] % d != 0:
            Q1 = Q1.multiply_variable(d / GCD(d, Q1[i, 0] // 2), i)

        ## Now perform the (symmetric) elementary operations to cancel out the (i,0) entries/
        Q1 = Q1.add_symmetric(-(Q1[i, 0] / 2) / (GCD(d, Q1[i, 0] // 2)), i, 0)

    ## Check that we're done!
    done_flag = True
    for i in range(1, n):
        if Q1[0, i] != 0:
            done_flag = False

    if not done_flag:
        raise RuntimeError(
            "There is a problem cancelling out the matrix entries! =O")

    ## Return the complementary matrix
    return Q1.extract_variables(range(1, n))
def complementary_subform_to_vector(self, v):
    """
    Finds the `(n-1)`-dim'l quadratic form orthogonal to the vector `v`.

    Note: This is usually not a direct summand!

    Technical Notes: There is a minor difference in the cancellation
    code here (form the C++ version) since the notation Q `[i,j]` indexes
    coefficients of the quadratic polynomial here, not the symmetric
    matrix.  Also, it produces a better splitting now, for the full
    lattice (as opposed to a sublattice in the C++ code) since we
    now extend `v` to a unimodular matrix.

    INPUT:
        `v` -- a list of self.dim() integers

    OUTPUT:
        a QuadraticForm over `ZZ`


    EXAMPLES::

        sage: Q1 = DiagonalQuadraticForm(ZZ, [1,3,5,7])
        sage: Q1.complementary_subform_to_vector([1,0,0,0])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 3 0 0 ]
        [ * 5 0 ]
        [ * * 7 ]

    ::

        sage: Q1.complementary_subform_to_vector([1,1,0,0])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 12 0 0 ]
        [ * 5 0 ]
        [ * * 7 ]

    ::

        sage: Q1.complementary_subform_to_vector([1,1,1,1])
        Quadratic form in 3 variables over Integer Ring with coefficients:
        [ 624 -480 -672 ]
        [ * 880 -1120 ]
        [ * * 1008 ]

    """
    n = self.dim()

    ## Copy the quadratic form
    Q = deepcopy(self)

    ## Find the first non-zero component of v, and call it nz  (Note: 0 <= nz < n)
    nz = 0
    while (nz < n) and (v[nz] == 0):
        nz += 1

    ## Abort if v is the zero vector
    if nz == n:
        raise TypeError("Oops, v cannot be the zero vector! =(")

    ## Make the change of basis matrix
    new_basis = extend_to_primitive(matrix(ZZ,n,1,v))

    ## Change Q (to Q1) to have v as its nz-th basis vector
    Q1 = Q(new_basis)

    ## Pick out the value Q(v) of the vector
    d = Q1[0, 0]

    #print Q1

    ## For each row/column, perform elementary operations to cancel them out.
    for i in range(1,n):

        ## Check if the (i,0)-entry is divisible by d,
        ## and stretch its row/column if not.
        if Q1[i,0] % d != 0:
            Q1 = Q1.multiply_variable(d / GCD(d, Q1[i, 0]//2), i)

        #print "After scaling at i =", i
        #print Q1

        ## Now perform the (symmetric) elementary operations to cancel out the (i,0) entries/
        Q1 = Q1.add_symmetric(-(Q1[i,0]/2) / (GCD(d, Q1[i,0]//2)), i, 0)

        #print "After cancelling at i =", i
        #print Q1

    ## Check that we're done!
    done_flag = True
    for i in range(1, n):
        if Q1[0,i] != 0:
            done_flag = False

    if done_flag == False:
        raise RuntimeError("There is a problem cancelling out the matrix entries! =O")


    ## Return the complementary matrix
    return Q1.extract_variables(range(1,n))
예제 #3
0
def find_p_neighbor_from_vec(self, p, v):
    """
    Finds the `p`-neighbor of this quadratic form associated to a given
    vector `v` satisfying:

    #. `Q(v) = 0  \pmod p`
    #. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.

    Reference:  Gonzalo Tornaria's Thesis, Thrm 3.5, p34.

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
        sage: v = vector([0,2,1,1])
        sage: X = Q.find_p_neighbor_from_vec(3,v); X
        Quadratic form in 4 variables over Integer Ring with coefficients:
        [ 3 10 0 -4 ]
        [ * 9 0 -6 ]
        [ * * 1 0 ]
        [ * * * 2 ]

    """
    R = self.base_ring()
    n = self.dim()
    B2 = self.matrix()

    ## Find a (dual) vector w with B(v,w) != 0 (mod p)
    v_dual = B2 * vector(v)     ## We want the dot product with this to not be divisible by 2*p.
    y_ind = 0
    while ((y_ind < n) and (v_dual[y_ind] % p) == 0):   ## Check the dot product for the std basis vectors!
        y_ind += 1
    if y_ind == n:
        raise RuntimeError("Oops!  One of the standard basis vectors should have worked.")
    w = vector([R(i == y_ind)  for i in range(n)])
    vw_prod = (v * self.matrix()).dot_product(w)

    ## DIAGNOSTIC
    #if vw_prod == 0:
    #   print "v = ", v
    #   print "v_dual = ", v_dual
    #   print "v_dual[y_ind] = ", v_dual[y_ind]
    #   print "(v_dual[y_ind] % p) = ", (v_dual[y_ind] % p)
    #   print "w = ", w
    #   print "p = ", p
    #   print "vw_prod = ", vw_prod
    #   raise RuntimeError, "ERROR: Why is vw_prod = 0?"

    ## DIAGNOSTIC
    #print "v = ", v
    #print "w = ", w
    #print "vw_prod = ", vw_prod


    ## Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
    s = self(v)
    if (s % p**2 != 0):
        al = (-s / (p * vw_prod)) % p
        v1 = v + p * al * w
        v1w_prod = (v1 * self.matrix()).dot_product(w)
    else:
        v1 = v
        v1w_prod = vw_prod

    ## DIAGNOSTIC
    #if (s % p**2 != 0):
    #    print "al = ", al
    #print "v1 = ", v1
    #print "v1w_prod = ", v1w_prod


    ## Construct a special p-divisible basis to use for the p-neighbor switch
    good_basis = extend_to_primitive([v1, w])
    for i in range(2,n):
        ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
        c = (ith_prod / v1w_prod) % p
        good_basis[i] = good_basis[i] - c * w  ## Ensures that this extension has <v_i, v> = 0 (mod p)

    ## DIAGNOSTIC
    #print "original good_basis = ", good_basis

    ## Perform the p-neighbor switch
    good_basis[0]  = vector([x/p  for x in good_basis[0]])    ## Divide v1 by p
    good_basis[1]  = good_basis[1] * p                          ## Multiply w by p

    ## Return the associated quadratic form
    M = matrix(good_basis)
    new_Q = deepcopy(self)                        ## Note: This avoids a circular import of QuadraticForm!
    new_Q.__init__(R, M * self.matrix() * M.transpose())
    return new_Q
    return QuadraticForm(R, M * self.matrix() * M.transpose())
def find_p_neighbor_from_vec(self, p, v):
    """
    Finds the `p`-neighbor of this quadratic form associated to a given
    vector `v` satisfying:

    #. `Q(v) = 0  \pmod p`
    #. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.

    Reference:  Gonzalo Tornaria's Thesis, Thrm 3.5, p34.

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
        sage: v = vector([0,2,1,1])
        sage: X = Q.find_p_neighbor_from_vec(3,v); X
        Quadratic form in 4 variables over Integer Ring with coefficients:
        [ 3 10 0 -4 ]
        [ * 9 0 -6 ]
        [ * * 1 0 ]
        [ * * * 2 ]

    """
    R = self.base_ring()
    n = self.dim()
    B2 = self.matrix()

    ## Find a (dual) vector w with B(v,w) != 0 (mod p)
    v_dual = B2 * vector(
        v)  ## We want the dot product with this to not be divisible by 2*p.
    y_ind = 0
    while ((y_ind < n) and (v_dual[y_ind] % p)
           == 0):  ## Check the dot product for the std basis vectors!
        y_ind += 1
    if y_ind == n:
        raise RuntimeError(
            "Oops!  One of the standard basis vectors should have worked.")
    w = vector([R(i == y_ind) for i in range(n)])
    vw_prod = (v * self.matrix()).dot_product(w)

    ## DIAGNOSTIC
    #if vw_prod == 0:
    #   print "v = ", v
    #   print "v_dual = ", v_dual
    #   print "v_dual[y_ind] = ", v_dual[y_ind]
    #   print "(v_dual[y_ind] % p) = ", (v_dual[y_ind] % p)
    #   print "w = ", w
    #   print "p = ", p
    #   print "vw_prod = ", vw_prod
    #   raise RuntimeError, "ERROR: Why is vw_prod = 0?"

    ## DIAGNOSTIC
    #print "v = ", v
    #print "w = ", w
    #print "vw_prod = ", vw_prod

    ## Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
    s = self(v)
    if (s % p**2 != 0):
        al = (-s / (p * vw_prod)) % p
        v1 = v + p * al * w
        v1w_prod = (v1 * self.matrix()).dot_product(w)
    else:
        v1 = v
        v1w_prod = vw_prod

    ## DIAGNOSTIC
    #if (s % p**2 != 0):
    #    print "al = ", al
    #print "v1 = ", v1
    #print "v1w_prod = ", v1w_prod

    ## Construct a special p-divisible basis to use for the p-neighbor switch
    good_basis = extend_to_primitive([v1, w])
    for i in range(2, n):
        ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
        c = (ith_prod / v1w_prod) % p
        good_basis[i] = good_basis[
            i] - c * w  ## Ensures that this extension has <v_i, v> = 0 (mod p)

    ## DIAGNOSTIC
    #print "original good_basis = ", good_basis

    ## Perform the p-neighbor switch
    good_basis[0] = vector([x / p for x in good_basis[0]])  ## Divide v1 by p
    good_basis[1] = good_basis[1] * p  ## Multiply w by p

    ## Return the associated quadratic form
    M = matrix(good_basis)
    new_Q = deepcopy(
        self)  ## Note: This avoids a circular import of QuadraticForm!
    new_Q.__init__(R, M * self.matrix() * M.transpose())
    return new_Q
    return QuadraticForm(R, M * self.matrix() * M.transpose())
def find_p_neighbor_from_vec(self, p, v):
    r"""
    Find the `p`-neighbor of this quadratic form associated to a given
    vector `v` satisfying:

    #. `Q(v) = 0  \pmod p`
    #. `v` is a non-singular point of the conic `Q(v) = 0 \pmod p`.

    Reference:  Gonzalo Tornaria's Thesis, Theorem 3.5, p34.

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ,[1,1,1,1])
        sage: v = vector([0,2,1,1])
        sage: X = Q.find_p_neighbor_from_vec(3,v); X
        Quadratic form in 4 variables over Integer Ring with coefficients:
        [ 3 10 0 -4 ]
        [ * 9 0 -6 ]
        [ * * 1 0 ]
        [ * * * 2 ]
    """
    R = self.base_ring()
    n = self.dim()
    B2 = self.matrix()

    # Find a (dual) vector w with B(v,w) != 0 (mod p)
    # We want the dot product with this to not be divisible by 2*p.
    v_dual = B2 * vector(v)

    y_ind = 0
    while ((y_ind < n) and (v_dual[y_ind] % p) == 0):
        # Check the dot product for the std basis vectors!
        y_ind += 1
    if y_ind == n:
        raise RuntimeError("Oops!  One of the standard basis vectors "
                           "should have worked.")
    w = vector(R, [R(i == y_ind) for i in range(n)])
    vw_prod = (v * self.matrix()).dot_product(w)

    # Lift the vector v to a vector v1 s.t. Q(v1) = 0 (mod p^2)
    s = self(v)
    if (s % p**2 != 0):
        al = (-s / (p * vw_prod)) % p
        v1 = v + p * al * w
        v1w_prod = (v1 * self.matrix()).dot_product(w)
    else:
        v1 = v
        v1w_prod = vw_prod

    # Construct a special p-divisible basis to use for the p-neighbor switch
    good_basis = extend_to_primitive([v1, w])
    for i in range(2, n):
        ith_prod = (good_basis[i] * self.matrix()).dot_product(v)
        c = (ith_prod / v1w_prod) % p
        good_basis[i] = good_basis[i] - c * w
        # Ensures that this extension has <v_i, v> = 0 (mod p)

    # Perform the p-neighbor switch
    good_basis[0] = vector([x / p for x in good_basis[0]])  # Divide v1 by p
    good_basis[1] = good_basis[1] * p  # Multiply w by p

    # Return the associated quadratic form
    M = matrix(good_basis)
    QF = self.parent()
    return QF(R, M * self.matrix() * M.transpose())