Python polytools.degree函数代码示例

def _invertx(p, x):
    """
    Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))``
    """
    p1 = poly_from_expr(p, x)[0]

    n = degree(p1)
    a = [c * x**(n - i) for (i,), c in p1.terms()]
    return Add(*a)

def _muly(p, x, y):
    """
    Returns ``_mexpand(y**deg*p.subs({x:x / y}))``
    """
    p1 = poly_from_expr(p, x)[0]

    n = degree(p1)
    a = [c * x**i * y**(n - i) for (i,), c in p1.terms()]
    return Add(*a)

def find_degree(M,s):
    '''
    Computes the degree of a polynomial matrix
    M : polynomial Matrix
    s: variable of the polynomial matrix
    
    Example:  TODO
    s=symbols('s')
    T=Matrix([[1, s**2, 0], [0, s, 1]])
    find_degree(T)
    '''
    return max(degree(poly,s) for poly in M if poly!=0 ) # for 0?

def make_rational_poly_simplify(var="x"):
    """
    Generates a rational expression of 4 polynomials, to be simplified.
    Example:
        ( (x**2 + 16*x + 60) / (x**2 - 36)) / 
        ( (x**2 - 2*x - 63) / (x**2 - 5*x - 36)

    x : charector for the variable to be solved for. defaults to random selection
        from the global list `alpha`.
                            OR
        a list of possible charectors. A random selection will be made from them.
    """
    if not var:
        var = random.choice(alpha)
    elif isinstance(var, list):
        var = random.choice(var)

    exclude = [var.upper(), var.lower()]
    x = sympy.Symbol(var)
    select = shuffle(range(-10,-1) + range(1,10))[:6]
    e1 = sympy.prod([x - i for i in shuffle(select)[:2]]).expand()
    e2 = sympy.prod([x - i for i in shuffle(select)[:2]]).expand()
    e3 = sympy.prod([x - i for i in shuffle(select)[:2]]).expand()
    e4 = sympy.prod([x - i for i in shuffle(select)[:2]]).expand()
    L = len(set([e1, e2, e3, e4]))
    e = ((e1/e2) / (e3 / e4))
    s1 = ''.join(["\\frac{", sympy.latex(e1), "}", "{", sympy.latex(e2), "}"])
    s2 = ''.join(["\\frac{", sympy.latex(e3), "}", "{", sympy.latex(e4), "}"])
    s3 = ''.join(["$$\\frac{", s1, "}", "{", s2, "}$$"])
    pieces = str(e.factor()).split("/")
    try:
        num, denom= [parse_expr(i).expand() for i in pieces]
    except:
        return make_rational_poly_simplify(var)
    if len(pieces) !=2 or L < 4 or degree(num) > 2 or  degree(denom) > 2:
        return make_rational_poly_simplify(var)
    return s3, render(num / denom)

def periodicity(f, symbol, check=False):
    """
    Tests the given function for periodicity in the given symbol.

    Parameters
    ==========

    f : Expr.
        The concerned function.
    symbol : Symbol
        The variable for which the period is to be determined.
    check : Boolean
        The flag to verify whether the value being returned is a period or not.

    Returns
    =======

    period
        The period of the function is returned.
        `None` is returned when the function is aperiodic or has a complex period.
        The value of `0` is returned as the period of a constant function.

    Raises
    ======

    NotImplementedError
        The value of the period computed cannot be verified.


    Notes
    =====

    Currently, we do not support functions with a complex period.
    The period of functions having complex periodic values such
    as `exp`, `sinh` is evaluated to `None`.

    The value returned might not be the "fundamental" period of the given
    function i.e. it may not be the smallest periodic value of the function.

    The verification of the period through the `check` flag is not reliable
    due to internal simplification of the given expression. Hence, it is set
    to `False` by default.

    Examples
    ========
    >>> from sympy import Symbol, sin, cos, tan, exp
    >>> from sympy.calculus.util import periodicity
    >>> x = Symbol('x')
    >>> f = sin(x) + sin(2*x) + sin(3*x)
    >>> periodicity(f, x)
    2*pi
    >>> periodicity(sin(x)*cos(x), x)
    pi
    >>> periodicity(exp(tan(2*x) - 1), x)
    pi/2
    >>> periodicity(sin(4*x)**cos(2*x), x)
    pi
    >>> periodicity(exp(x), x)

    """
    from sympy import simplify, lcm_list
    from sympy.functions.elementary.complexes import Abs
    from sympy.functions.elementary.trigonometric import (
        TrigonometricFunction, sin, cos, csc, sec)
    from sympy.solvers.decompogen import decompogen
    from sympy.core import Mod
    from sympy.polys.polytools import degree
    from sympy.core.function import diff
    from sympy.core.relational import Relational

    def _check(orig_f, period):
        '''Return the checked period or raise an error.'''
        new_f = orig_f.subs(symbol, symbol + period)
        if new_f.equals(orig_f):
            return period
        else:
            raise NotImplementedError(filldedent('''
                The period of the given function cannot be verified.
                When `%s` was replaced with `%s + %s` in `%s`, the result
                was `%s` which was not recognized as being the same as
                the original function.
                So either the period was wrong or the two forms were
                not recognized as being equal.
                Set check=False to obtain the value.''' %
                (symbol, symbol, period, orig_f, new_f)))

    orig_f = f
    f = simplify(orig_f)
    period = None

    if symbol not in f.free_symbols:
        return S.Zero

    if isinstance(f, Relational):
        f = f.lhs - f.rhs

    if isinstance(f, TrigonometricFunction):
        try:
            period = f.period(symbol)
        except NotImplementedError:
#.........这里部分代码省略.........

def test_minpoly_compose():
    # issue 6868
    eq = S('''
        -1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
        sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
        sqrt(15)*I/28800000)**(1/3)))''')
    mp = minimal_polynomial(eq + 3, x)
    assert mp == 8000*x**2 - 48000*x + 71999

    # issue 5888
    assert minimal_polynomial(exp(I*pi/8), x) == x**8 + 1

    mp = minimal_polynomial(sin(pi/7) + sqrt(2), x)
    assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
        770912*x**4 - 268432*x**2 + 28561
    mp = minimal_polynomial(cos(pi/7) + sqrt(2), x)
    assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
            232*x - 239
    mp = minimal_polynomial(exp(I*pi/7) + sqrt(2), x)
    assert mp == x**12 - 2*x**11 - 9*x**10 + 16*x**9 + 43*x**8 - 70*x**7 - 97*x**6 + 126*x**5 + 211*x**4 - 212*x**3 - 37*x**2 + 142*x + 127

    mp = minimal_polynomial(sin(pi/7) + sqrt(2), x)
    assert mp == 4096*x**12 - 63488*x**10 + 351488*x**8 - 826496*x**6 + \
        770912*x**4 - 268432*x**2 + 28561
    mp = minimal_polynomial(cos(pi/7) + sqrt(2), x)
    assert mp == 64*x**6 - 64*x**5 - 432*x**4 + 304*x**3 + 712*x**2 - \
            232*x - 239
    mp = minimal_polynomial(exp(I*pi/7) + sqrt(2), x)
    assert mp == x**12 - 2*x**11 - 9*x**10 + 16*x**9 + 43*x**8 - 70*x**7 - 97*x**6 + 126*x**5 + 211*x**4 - 212*x**3 - 37*x**2 + 142*x + 127

    mp = minimal_polynomial(exp(2*I*pi/7), x)
    assert mp == x**6 + x**5 + x**4 + x**3 + x**2 + x + 1
    mp = minimal_polynomial(exp(2*I*pi/15), x)
    assert mp == x**8 - x**7 + x**5 - x**4 + x**3 - x + 1
    mp = minimal_polynomial(cos(2*pi/7), x)
    assert mp == 8*x**3 + 4*x**2 - 4*x - 1
    mp = minimal_polynomial(sin(2*pi/7), x)
    ex = (5*cos(2*pi/7) - 7)/(9*cos(pi/7) - 5*cos(3*pi/7))
    mp = minimal_polynomial(ex, x)
    assert mp == x**3 + 2*x**2 - x - 1
    assert minimal_polynomial(-1/(2*cos(pi/7)), x) == x**3 + 2*x**2 - x - 1
    assert minimal_polynomial(sin(2*pi/15), x) == \
            256*x**8 - 448*x**6 + 224*x**4 - 32*x**2 + 1
    assert minimal_polynomial(sin(5*pi/14), x) == 8*x**3 - 4*x**2 - 4*x + 1
    assert minimal_polynomial(cos(pi/15), x) == 16*x**4 + 8*x**3 - 16*x**2 - 8*x + 1

    ex = rootof(x**3 +x*4 + 1, 0)
    mp = minimal_polynomial(ex, x)
    assert mp == x**3 + 4*x + 1
    mp = minimal_polynomial(ex + 1, x)
    assert mp == x**3 - 3*x**2 + 7*x - 4
    assert minimal_polynomial(exp(I*pi/3), x) == x**2 - x + 1
    assert minimal_polynomial(exp(I*pi/4), x) == x**4 + 1
    assert minimal_polynomial(exp(I*pi/6), x) == x**4 - x**2 + 1
    assert minimal_polynomial(exp(I*pi/9), x) == x**6 - x**3 + 1
    assert minimal_polynomial(exp(I*pi/10), x) == x**8 - x**6 + x**4 - x**2 + 1
    assert minimal_polynomial(sin(pi/9), x) == 64*x**6 - 96*x**4 + 36*x**2 - 3
    assert minimal_polynomial(sin(pi/11), x) == 1024*x**10 - 2816*x**8 + \
            2816*x**6 - 1232*x**4 + 220*x**2 - 11

    ex = 2**Rational(1, 3)*exp(Rational(2, 3)*I*pi)
    assert minimal_polynomial(ex, x) == x**3 - 2

    raises(NotAlgebraic, lambda: minimal_polynomial(cos(pi*sqrt(2)), x))
    raises(NotAlgebraic, lambda: minimal_polynomial(sin(pi*sqrt(2)), x))
    raises(NotAlgebraic, lambda: minimal_polynomial(exp(I*pi*sqrt(2)), x))

    # issue 5934
    ex = 1/(-36000 - 7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
        24*sqrt(10)*sqrt(-sqrt(5) + 5))**2) + 1
    raises(ZeroDivisionError, lambda: minimal_polynomial(ex, x))

    ex = sqrt(1 + 2**Rational(1,3)) + sqrt(1 + 2**Rational(1,4)) + sqrt(2)
    mp = minimal_polynomial(ex, x)
    assert degree(mp) == 48 and mp.subs({x:0}) == -16630256576

def _minpoly_groebner(ex, x, cls):
    """
    Computes the minimal polynomial of an algebraic number
    using Groebner bases

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, Rational
    >>> from sympy.abc import x
    >>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False)
    x**2 - 2*x - 1

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = next(generator)
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base**(-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (
                        ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base**exp

                if expr not in mapping:
                    return update_mapping(expr, 1/exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1/ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
            a = []
            for p in ex.args:
                if p.is_Add:
                    return False
                if p.is_Pow:
                    if p.base.is_Add and p.exp > 0:
                        return False

            if hit:
#.........这里部分代码省略.........

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic element.

    Parameters
    ==========

    ex : algebraic element expression
    x : independent variable of the minimal polynomial

    Options
    =======

    compose : if ``True`` ``_minpoly_compose`` is used, if ``False`` the ``groebner`` algorithm
    polys : if ``True`` returns a ``Poly`` object
    domain : ground domain

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    If no ground domain is given, it will be generated automatically from the expression.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve, QQ
    >>> from sympy.abc import x, y

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
    x - sqrt(2)
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3
    >>> minimal_polynomial(sqrt(y), x)
    x**2 - y

    """
    from sympy.polys.polytools import degree
    from sympy.polys.domains import FractionField
    from sympy.core.basic import preorder_traversal

    compose = args.get('compose', True)
    polys = args.get('polys', False)
    dom = args.get('domain', None)

    ex = sympify(ex)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if not dom:
        dom = FractionField(QQ, list(ex.free_symbols)) if ex.free_symbols else QQ
    if hasattr(dom, 'symbols') and x in dom.symbols:
        raise GeneratorsError("the variable %s is an element of the ground domain %s" % (x, dom))

    if compose:
        result = _minpoly_compose(ex, x, dom)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c.is_negative:
            result = expand_mul(-result)
        return cls(result, x, field=True) if polys else result.collect(x)

    if not dom.is_QQ:
        raise NotImplementedError("groebner method only works for QQ")

    result = _minpoly_groebner(ex, x, cls)
    return cls(result, x, field=True) if polys else result.collect(x)

def _minpoly_op_algebraic_element(op, ex1, ex2, x, dom, mp1=None, mp2=None):
    """
    return the minimal polynomial for ``op(ex1, ex2)``

    Parameters
    ==========

    op : operation ``Add`` or ``Mul``
    ex1, ex2 : expressions for the algebraic elements
    x : indeterminate of the polynomials
    dom: ground domain
    mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None

    Examples
    ========

    >>> from sympy import sqrt, Add, Mul, QQ
    >>> from sympy.polys.numberfields import _minpoly_op_algebraic_element
    >>> from sympy.abc import x, y
    >>> p1 = sqrt(sqrt(2) + 1)
    >>> p2 = sqrt(sqrt(2) - 1)
    >>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ)
    x - 1
    >>> q1 = sqrt(y)
    >>> q2 = 1 / y
    >>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y))
    x**2*y**2 - 2*x*y - y**3 + 1

    References
    ==========

    [1] http://en.wikipedia.org/wiki/Resultant
    [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
    "Degrees of sums in a separable field extension".
    """
    from sympy import gcd
    y = Dummy(str(x))
    if mp1 is None:
        mp1 = _minpoly_compose(ex1, x, dom)
    if mp2 is None:
        mp2 = _minpoly_compose(ex2, y, dom)
    else:
        mp2 = mp2.subs({x: y})

    if op is Add:
        # mp1a = mp1.subs({x: x - y})
        (p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y)
        r = p1.compose(p2)
        mp1a = r.as_expr()
    elif op is Mul:
        mp1a = _muly(mp1, x, y)
    else:
        raise NotImplementedError('option not available')

    r = resultant(mp1a, mp2, gens=[y, x])

    deg1 = degree(mp1, x)
    deg2 = degree(mp2, y)
    if op is Add and gcd(deg1, deg2) == 1:
        # `r` is irreducible, see [2]
        return r
    if op is Mul and deg1 == 1 or deg2 == 1:
        # if deg1 = 1, then mp1 = x - a; mp1a = x - y - a;
        # r = mp2(x - a), so that `r` is irreducible
        return r

    r = Poly(r, x, domain=dom)
    _, factors = r.factor_list()
    res = _choose_factor(factors, x, op(ex1, ex2), dom)
    return res.as_expr()

def row_degrees(A,s):
    """
    return a list of the degrees of the row of matrix
    """
    return [max(degree(poly,s)  for poly in A[i,:] if poly!=0 ) for i in range(A.rows)]

def _parts_rule(integrand, symbol):
    # LIATE rule:
    # log, inverse trig, algebraic, trigonometric, exponential
    def pull_out_algebraic(integrand):
        integrand = integrand.cancel().together()
        algebraic = [arg for arg in integrand.args if arg.is_algebraic_expr(symbol)]
        if algebraic:
            u = sympy.Mul(*algebraic)
            dv = (integrand / u).cancel()
            return u, dv

    def pull_out_u(*functions):
        def pull_out_u_rl(integrand):
            if any([integrand.has(f) for f in functions]):
                args = [arg for arg in integrand.args
                        if any(isinstance(arg, cls) for cls in functions)]
                if args:
                    u = reduce(lambda a,b: a*b, args)
                    dv = integrand / u
                    return u, dv

        return pull_out_u_rl

    liate_rules = [pull_out_u(sympy.log), pull_out_u(sympy.atan, sympy.asin, sympy.acos),
                   pull_out_algebraic, pull_out_u(sympy.sin, sympy.cos),
                   pull_out_u(sympy.exp)]


    dummy = sympy.Dummy("temporary")
    # we can integrate log(x) and atan(x) by setting dv = 1
    if isinstance(integrand, (sympy.log, sympy.atan, sympy.asin, sympy.acos)):
        integrand = dummy * integrand

    for index, rule in enumerate(liate_rules):
        result = rule(integrand)

        if result:
            u, dv = result

            # Don't pick u to be a constant if possible
            if symbol not in u.free_symbols and not u.has(dummy):
                return

            u = u.subs(dummy, 1)
            dv = dv.subs(dummy, 1)

            # Don't pick a non-polynomial algebraic to be differentiated
            if rule == pull_out_algebraic and not u.is_polynomial(symbol):
                return
            # Don't trade one logarithm for another
            if isinstance(u, sympy.log):
                rec_dv = 1/dv
                if (rec_dv.is_polynomial(symbol) and
                    degree(rec_dv, symbol) == 1):
                        return

            for rule in liate_rules[index + 1:]:
                r = rule(integrand)
                # make sure dv is amenable to integration
                if r and r[0].subs(dummy, 1).equals(dv):
                    du = u.diff(symbol)
                    v_step = integral_steps(sympy.simplify(dv), symbol)
                    v = _manualintegrate(v_step)

                    return u, dv, v, du, v_step

def minimal_polynomial(ex, x=None, compose=True, polys=False, domain=None):
    """
    Computes the minimal polynomial of an algebraic element.

    Parameters
    ==========

    ex : Expr
        Element or expression whose minimal polynomial is to be calculated.

    x : Symbol, optional
        Independent variable of the minimal polynomial

    compose : boolean, optional (default=True)
        Method to use for computing minimal polynomial. If ``compose=True``
        (default) then ``_minpoly_compose`` is used, if ``compose=False`` then
        groebner bases are used.

    polys : boolean, optional (default=False)
        If ``True`` returns a ``Poly`` object else an ``Expr`` object.

    domain : Domain, optional
        Ground domain

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    If no ground domain is given, it will be generated automatically from the expression.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve, QQ
    >>> from sympy.abc import x, y

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
    x - sqrt(2)
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3
    >>> minimal_polynomial(sqrt(y), x)
    x**2 - y

    """
    from sympy.polys.polytools import degree
    from sympy.polys.domains import FractionField
    from sympy.core.basic import preorder_traversal

    ex = sympify(ex)
    if ex.is_number:
        # not sure if it's always needed but try it for numbers (issue 8354)
        ex = _mexpand(ex, recursive=True)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if not domain:
        if ex.free_symbols:
            domain = FractionField(QQ, list(ex.free_symbols))
        else:
            domain = QQ
    if hasattr(domain, 'symbols') and x in domain.symbols:
        raise GeneratorsError("the variable %s is an element of the ground "
                              "domain %s" % (x, domain))

    if compose:
        result = _minpoly_compose(ex, x, domain)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c.is_negative:
            result = expand_mul(-result)
        return cls(result, x, field=True) if polys else result.collect(x)

    if not domain.is_QQ:
        raise NotImplementedError("groebner method only works for QQ")

    result = _minpoly_groebner(ex, x, cls)
    return cls(result, x, field=True) if polys else result.collect(x)

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    Parameters
    ==========

    ex : algebraic number expression

    x : indipendent variable of the minimal polynomial

    Options
    =======

    compose : if ``True`` _minpoly1`` is used, else the ``groebner`` algorithm

    polys : if ``True`` returns a ``Poly`` object

    Notes
    =====

    By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
    are computed, then the arithmetic operations on them are performed using the resultant
    and factorization.
    If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
    The default algorithm stalls less frequently.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt, solve
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1
    >>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
    x**3 + x + 3

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_multinomial
    from sympy.core.basic import preorder_traversal

    compose = args.get('compose', True)
    polys = args.get('polys', False)
    ex = sympify(ex)
    for expr in preorder_traversal(ex):
        if expr.is_AlgebraicNumber:
            compose = False
            break

    if ex.is_AlgebraicNumber:
        compose = False

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy('x'), PurePoly

    if compose:
        result = _minpoly1(ex, x)
        result = result.primitive()[1]
        c = result.coeff(x**degree(result, x))
        if c < 0:
            result = expand_mul(-result)
            c = -c
        return cls(result, x, field=True) if polys else result

    generator = numbered_symbols('a', cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a**exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Mul:
            return Mul(*[ bottom_up_scan(g) for g in ex.args ])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
#.........这里部分代码省略.........

def _minpoly_op_algebraic_number(ex1, ex2, x, mp1=None, mp2=None, op=Add):
    """
    return the minimal polinomial for ``op(ex1, ex2)``

    Parameters
    ==========

    ex1, ex2 : expressions for the algebraic numbers
    x : indeterminate of the polynomials
    mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None
    op : operation ``Add`` or ``Mul``

    Examples
    ========

    >>> from sympy import sqrt, Mul
    >>> from sympy.polys.numberfields import _minpoly_op_algebraic_number
    >>> from sympy.abc import x
    >>> p1 = sqrt(sqrt(2) + 1)
    >>> p2 = sqrt(sqrt(2) - 1)
    >>> _minpoly_op_algebraic_number(p1, p2, x, op=Mul)
    x - 1

    References
    ==========

    [1] http://en.wikipedia.org/wiki/Resultant
    [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
    "Degrees of sums in a separable field extension".
    """
    from sympy import gcd
    y = Dummy(str(x))
    if mp1 is None:
        mp1 = _minpoly1(ex1, x)
    if mp2 is None:
        mp2 = _minpoly1(ex2, y)
    else:
        mp2 = mp2.subs({x:y})

    if op is Add:
        # mp1a = mp1.subs({x:x - y})
        (p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y)
        r = p1.compose(p2)
        mp1a = r.as_expr()
    elif op is Mul:
        mp1a = _muly(mp1, x, y)
    else:
        raise NotImplementedError('option not available')
    r = resultant(mp1a, mp2, gens=[y, x])

    deg1 = degree(mp1, x)
    deg2 = degree(mp2, y)
    if op is Add and gcd(deg1, deg2) == 1:
        # `r` is irreducible, see [2]
        return r
    if op is Mul and deg1 == 1 or deg2 == 1:
        # if deg1 = 1, then mp1 = x - a; mp1a = x - y - a;
        # r = mp2(x - a), so that `r` is irreducible
        return r
    _, factors = factor_list(r)
    if op in [Add, Mul]:
        ex = op(ex1, ex2)
    res = _choose_factor(factors, x, ex)
    return res

def minimal_polynomial(ex, x=None, **args):
    """
    Computes the minimal polynomial of an algebraic number.

    Examples
    ========

    >>> from sympy import minimal_polynomial, sqrt
    >>> from sympy.abc import x

    >>> minimal_polynomial(sqrt(2), x)
    x**2 - 2
    >>> minimal_polynomial(sqrt(2) + sqrt(3), x)
    x**4 - 10*x**2 + 1

    """
    from sympy.polys.polytools import degree
    from sympy.core.function import expand_mul, expand_multinomial
    from sympy.simplify.simplify import _is_sum_surds

    generator = numbered_symbols("a", cls=Dummy)
    mapping, symbols, replace = {}, {}, []

    ex = sympify(ex)

    if x is not None:
        x, cls = sympify(x), Poly
    else:
        x, cls = Dummy("x"), PurePoly

    def update_mapping(ex, exp, base=None):
        a = generator.next()
        symbols[ex] = a

        if base is not None:
            mapping[ex] = a ** exp + base
        else:
            mapping[ex] = exp.as_expr(a)

        return a

    def bottom_up_scan(ex):
        if ex.is_Atom:
            if ex is S.ImaginaryUnit:
                if ex not in mapping:
                    return update_mapping(ex, 2, 1)
                else:
                    return symbols[ex]
            elif ex.is_Rational:
                return ex
        elif ex.is_Add:
            return Add(*[bottom_up_scan(g) for g in ex.args])
        elif ex.is_Mul:
            return Mul(*[bottom_up_scan(g) for g in ex.args])
        elif ex.is_Pow:
            if ex.exp.is_Rational:
                if ex.exp < 0 and ex.base.is_Add:
                    coeff, terms = ex.base.as_coeff_add()
                    elt, _ = primitive_element(terms, polys=True)

                    alg = ex.base - coeff

                    # XXX: turn this into eval()
                    inverse = invert(elt.gen + coeff, elt).as_expr()
                    base = inverse.subs(elt.gen, alg).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base)
                    else:
                        ex = base ** (-ex.exp)
                if not ex.exp.is_Integer:
                    base, exp = (ex.base ** ex.exp.p).expand(), Rational(1, ex.exp.q)
                else:
                    base, exp = ex.base, ex.exp
                base = bottom_up_scan(base)
                expr = base ** exp

                if expr not in mapping:
                    return update_mapping(expr, 1 / exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex.root not in mapping:
                return update_mapping(ex.root, ex.minpoly)
            else:
                return symbols[ex.root]

        raise NotAlgebraic("%s doesn't seem to be an algebraic number" % ex)

    def simpler_inverse(ex):
        """
        Returns True if it is more likely that the minimal polynomial
        algorithm works better with the inverse
        """
        if ex.is_Pow:
            if (1 / ex.exp).is_integer and ex.exp < 0:
                if ex.base.is_Add:
                    return True
        if ex.is_Mul:
            hit = True
#.........这里部分代码省略.........