(eq)
Helper routine used by diop_solve() to find the type of the ``eq`` etc.

    Returns a tuple containing the type of the diophantine equation along with
    the variables(free symbols) and their coefficients. Variables are returned
    as a list and coefficients are returned as a dict with the key being the
    respective term and the constant term is keyed to Integer(1). Type is an
    element in the set {"linear", "binary_quadratic", "general_pythagorean",
    "homogeneous_ternary_quadratic", "univariate", "general_sum_of_squares"}

    Usage
    =====

    ``classify_diop(eq)``: Return variables, coefficients and type of the
    ``eq``.

    Details
    =======

    ``eq`` should be an expression which is assumed to be zero.

    Examples
    ========

    >>> from sympy.solvers.diophantine import classify_diop
    >>> from sympy.abc import x, y, z, w, t
    >>> classify_diop(4*x + 6*y - 4)
    ([x, y], {1: -4, x: 4, y: 6}, 'linear')
    >>> classify_diop(x + 3*y -4*z + 5)
    ([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear')
    >>> classify_diop(x**2 + y**2 - x*y + x + 5)
    ([x, y], {1: 5, x: 1, x**2: 1, y: 0, y**2: 1, x*y: -1}, 'binary_quadratic')
Helper routine used by diop_solve() to find the type of the ``eq`` etc.
Helper routine used by diop_solve () to find the type of the eq etc .
def classify_diop(eq):
    """
    Helper routine used by diop_solve() to find the type of the ``eq`` etc.

    Returns a tuple containing the type of the diophantine equation along with
    the variables(free symbols) and their coefficients. Variables are returned
    as a list and coefficients are returned as a dict with the key being the
    respective term and the constant term is keyed to Integer(1). Type is an
    element in the set {"linear", "binary_quadratic", "general_pythagorean",
    "homogeneous_ternary_quadratic", "univariate", "general_sum_of_squares"}

    Usage
    =====

    ``classify_diop(eq)``: Return variables, coefficients and type of the
    ``eq``.

    Details
    =======

    ``eq`` should be an expression which is assumed to be zero.

    Examples
    ========

    >>> from sympy.solvers.diophantine import classify_diop
    >>> from sympy.abc import x, y, z, w, t
    >>> classify_diop(4*x + 6*y - 4)
    ([x, y], {1: -4, x: 4, y: 6}, 'linear')
    >>> classify_diop(x + 3*y -4*z + 5)
    ([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear')
    >>> classify_diop(x**2 + y**2 - x*y + x + 5)
    ([x, y], {1: 5, x: 1, x**2: 1, y: 0, y**2: 1, x*y: -1}, 'binary_quadratic')
    """
    eq = eq.expand(force=True)
    var = list(eq.free_symbols)
    var.sort(key=default_sort_key)

    coeff = {}
    diop_type = None

    coeff = dict([reversed(t.as_independent(*var)) for t in eq.args])
    for v in coeff:
        if not isinstance(coeff[v], Integer):
            raise TypeError("Coefficients should be Integers")

    if len(var) == 1:
        diop_type = "univariate"
    elif Poly(eq).total_degree() == 1:
        diop_type = "linear"
    elif Poly(eq).total_degree() == 2 and len(var) == 2:
        diop_type = "binary_quadratic"
        x, y = var[:2]

        if isinstance(eq, Mul):
            coeff = {x**2: 0, x*y: eq.args[0], y**2: 0, x: 0, y: 0, Integer(1): 0}
        else:
            for term in [x**2, y**2, x*y, x, y, Integer(1)]:
                if term not in coeff.keys():
                    coeff[term] = Integer(0)

    elif Poly(eq).total_degree() == 2 and len(var) == 3 and Integer(1) not in coeff.keys():
        for v in var:
            if v in coeff.keys():
                diop_type = "inhomogeneous_ternary_quadratic"
                break
        else:
            diop_type = "homogeneous_ternary_quadratic"

            x, y, z = var[:3]

            for term in [x**2, y**2, z**2, x*y, y*z, x*z]:
                if term not in coeff.keys():
                    coeff[term] = Integer(0)

    elif Poly(eq).degree() == 2 and len(var) >= 3:

        for v in var:
            if v in coeff.keys():
                diop_type = "inhomogeneous_general_quadratic"
                break

        else:
            if Integer(1) in coeff.keys():
                constant_term = True
            else:
                constant_term = False

            non_square_degree_2_terms = False
            for v in var:
                for u in var:
                    if u != v and u*v in coeff.keys():
                        non_square_degree_2_terms = True
                        break
                if non_square_degree_2_terms:
                    break

            if constant_term and non_square_degree_2_terms:
                diop_type = "inhomogeneous_general_quadratic"

            elif constant_term and not non_square_degree_2_terms:
                for v in var:
                    if coeff[v**2] != 1:
                        break
                else:
                    diop_type = "general_sum_of_squares"

            elif not constant_term and non_square_degree_2_terms:
                diop_type = "homogeneous_general_quadratic"

            else:
                coeff_sign_sum = 0

                for v in var:
                    if not isinstance(sqrt(abs(Integer(coeff[v**2]))), Integer):
                        break
                    coeff_sign_sum = coeff_sign_sum + sign(coeff[v**2])
                else:
                    if abs(coeff_sign_sum) == len(var) - 2 and not constant_term:
                        diop_type = "general_pythagorean"

    elif Poly(eq).total_degree() == 3 and len(var) == 2:

        x, y = var[:2]
        diop_type = "cubic_thue"

        for term in [x**3, x**2*y, x*y**2, y**3, Integer(1)]:
            if term not in coeff.keys():
                coeff[term] == Integer(0)

    if diop_type is not None:
        return var, coeff, diop_type
    else:
        raise NotImplementedError("Still not implemented")
def classify_diop ( eq ) : eq = eq . expand ( force = True ) var = list ( eq . free_symbols ) var . sort ( key = default_sort_key ) coeff = { } diop_type = None coeff = dict ( [ reversed ( t . as_independent ( * var ) ) for t in eq . args ] ) for v in coeff : if not isinstance ( coeff [ v ] , Integer ) : raise TypeError ( "Coefficients should be Integers" ) if len ( var ) == 1 : diop_type = "univariate" elif Poly ( eq ) . total_degree ( ) == 1 : diop_type = "linear" elif Poly ( eq ) . total_degree ( ) == 2 and len ( var ) == 2 : diop_type = "binary_quadratic" x , y = var [ : 2 ] if isinstance ( eq , Mul ) : coeff = { x ** 2 : 0 , x * y : eq . args [ 0 ] , y ** 2 : 0 , x : 0 , y : 0 , Integer ( 1 ) : 0 } else : for term in [ x ** 2 , y ** 2 , x * y , x , y , Integer ( 1 ) ] : if term not in coeff . keys ( ) : coeff [ term ] = Integer ( 0 ) elif Poly ( eq ) . total_degree ( ) == 2 and len ( var ) == 3 and Integer ( 1 ) not in coeff . keys ( ) : for v in var : if v in coeff . keys ( ) : diop_type = "inhomogeneous_ternary_quadratic" break else : diop_type = "homogeneous_ternary_quadratic" x , y , z = var [ : 3 ] for term in [ x ** 2 , y ** 2 , z ** 2 , x * y , y * z , x * z ] : if term not in coeff . keys ( ) : coeff [ term ] = Integer ( 0 ) elif Poly ( eq ) . degree ( ) == 2 and len ( var ) >= 3 : for v in var : if v in coeff . keys ( ) : diop_type = "inhomogeneous_general_quadratic" break else : if Integer ( 1 ) in coeff . keys ( ) : constant_term = True else : constant_term = False non_square_degree_2_terms = False for v in var : for u in var : if u != v and u * v in coeff . keys ( ) : non_square_degree_2_terms = True break if non_square_degree_2_terms : break if constant_term and non_square_degree_2_terms : diop_type = "inhomogeneous_general_quadratic" elif constant_term and not non_square_degree_2_terms : for v in var : if coeff [ v ** 2 ] != 1 : break else : diop_type = "general_sum_of_squares" elif not constant_term and non_square_degree_2_terms : diop_type = "homogeneous_general_quadratic" else : coeff_sign_sum = 0 for v in var : if not isinstance ( sqrt ( abs ( Integer ( coeff [ v ** 2 ] ) ) ) , Integer ) : break coeff_sign_sum = coeff_sign_sum + sign ( coeff [ v ** 2 ] ) else : if abs ( coeff_sign_sum ) == len ( var ) - 2 and not constant_term : diop_type = "general_pythagorean" elif Poly ( eq ) . total_degree ( ) == 3 and len ( var ) == 2 : x , y = var [ : 2 ] diop_type = "cubic_thue" for term in [ x ** 3 , x ** 2 * y , x * y ** 2 , y ** 3 , Integer ( 1 ) ] : if term not in coeff . keys ( ) : coeff [ term ] == Integer ( 0 ) if diop_type is not None : return var , coeff , diop_type else : raise NotImplementedError ( "Still not implemented" )