ai-content-maker/.venv/Lib/site-packages/sympy/polys/numberfields/minpoly.py

"""Minimal polynomials for algebraic numbers."""

from functools import reduce

from sympy.core.add import Add
from sympy.core.exprtools import Factors
from sympy.core.function import expand_mul, expand_multinomial, _mexpand
from sympy.core.mul import Mul
from sympy.core.numbers import (I, Rational, pi, _illegal)
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.core.traversal import preorder_traversal
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.trigonometric import cos, sin, tan
from sympy.ntheory.factor_ import divisors
from sympy.utilities.iterables import subsets

from sympy.polys.domains import ZZ, QQ, FractionField
from sympy.polys.orthopolys import dup_chebyshevt
from sympy.polys.polyerrors import (
    NotAlgebraic,
    GeneratorsError,
)
from sympy.polys.polytools import (
    Poly, PurePoly, invert, factor_list, groebner, resultant,
    degree, poly_from_expr, parallel_poly_from_expr, lcm
)
from sympy.polys.polyutils import dict_from_expr, expr_from_dict
from sympy.polys.ring_series import rs_compose_add
from sympy.polys.rings import ring
from sympy.polys.rootoftools import CRootOf
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.utilities import (
    numbered_symbols, public, sift
)


def _choose_factor(factors, x, v, dom=QQ, prec=200, bound=5):
    """
    Return a factor having root ``v``
    It is assumed that one of the factors has root ``v``.
    """

    if isinstance(factors[0], tuple):
        factors = [f[0] for f in factors]
    if len(factors) == 1:
        return factors[0]

    prec1 = 10
    points = {}
    symbols = dom.symbols if hasattr(dom, 'symbols') else []
    while prec1 <= prec:
        # when dealing with non-Rational numbers we usually evaluate
        # with `subs` argument but we only need a ballpark evaluation
        fe = [f.as_expr().xreplace({x:v}) for f in factors]
        if v.is_number:
            fe = [f.n(prec) for f in fe]

        # assign integers [0, n) to symbols (if any)
        for n in subsets(range(bound), k=len(symbols), repetition=True):
            for s, i in zip(symbols, n):
                points[s] = i

            # evaluate the expression at these points
            candidates = [(abs(f.subs(points).n(prec1)), i)
                for i,f in enumerate(fe)]

            # if we get invalid numbers (e.g. from division by zero)
            # we try again
            if any(i in _illegal for i, _ in candidates):
                continue

            # find the smallest two -- if they differ significantly
            # then we assume we have found the factor that becomes
            # 0 when v is substituted into it
            can = sorted(candidates)
            (a, ix), (b, _) = can[:2]
            if b > a * 10**6:  # XXX what to use?
                return factors[ix]

        prec1 *= 2

    raise NotImplementedError("multiple candidates for the minimal polynomial of %s" % v)


def _is_sum_surds(p):
    args = p.args if p.is_Add else [p]
    for y in args:
        if not ((y**2).is_Rational and y.is_extended_real):
            return False
    return True


def _separate_sq(p):
    """
    helper function for ``_minimal_polynomial_sq``

    It selects a rational ``g`` such that the polynomial ``p``
    consists of a sum of terms whose surds squared have gcd equal to ``g``
    and a sum of terms with surds squared prime with ``g``;
    then it takes the field norm to eliminate ``sqrt(g)``

    See simplify.simplify.split_surds and polytools.sqf_norm.

    Examples
    ========

    >>> from sympy import sqrt
    >>> from sympy.abc import x
    >>> from sympy.polys.numberfields.minpoly import _separate_sq
    >>> p= -x + sqrt(2) + sqrt(3) + sqrt(7)
    >>> p = _separate_sq(p); p
    -x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8
    >>> p = _separate_sq(p); p
    -x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20
    >>> p = _separate_sq(p); p
    -x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400

    """
    def is_sqrt(expr):
        return expr.is_Pow and expr.exp is S.Half
    # p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)]
    a = []
    for y in p.args:
        if not y.is_Mul:
            if is_sqrt(y):
                a.append((S.One, y**2))
            elif y.is_Atom:
                a.append((y, S.One))
            elif y.is_Pow and y.exp.is_integer:
                a.append((y, S.One))
            else:
                raise NotImplementedError
        else:
            T, F = sift(y.args, is_sqrt, binary=True)
            a.append((Mul(*F), Mul(*T)**2))
    a.sort(key=lambda z: z[1])
    if a[-1][1] is S.One:
        # there are no surds
        return p
    surds = [z for y, z in a]
    for i in range(len(surds)):
        if surds[i] != 1:
            break
    from sympy.simplify.radsimp import _split_gcd
    g, b1, b2 = _split_gcd(*surds[i:])
    a1 = []
    a2 = []
    for y, z in a:
        if z in b1:
            a1.append(y*z**S.Half)
        else:
            a2.append(y*z**S.Half)
    p1 = Add(*a1)
    p2 = Add(*a2)
    p = _mexpand(p1**2) - _mexpand(p2**2)
    return p

def _minimal_polynomial_sq(p, n, x):
    """
    Returns the minimal polynomial for the ``nth-root`` of a sum of surds
    or ``None`` if it fails.

    Parameters
    ==========

    p : sum of surds
    n : positive integer
    x : variable of the returned polynomial

    Examples
    ========

    >>> from sympy.polys.numberfields.minpoly import _minimal_polynomial_sq
    >>> from sympy import sqrt
    >>> from sympy.abc import x
    >>> q = 1 + sqrt(2) + sqrt(3)
    >>> _minimal_polynomial_sq(q, 3, x)
    x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8

    """
    p = sympify(p)
    n = sympify(n)
    if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
        return None
    pn = p**Rational(1, n)
    # eliminate the square roots
    p -= x
    while 1:
        p1 = _separate_sq(p)
        if p1 is p:
            p = p1.subs({x:x**n})
            break
        else:
            p = p1

    # _separate_sq eliminates field extensions in a minimal way, so that
    # if n = 1 then `p = constant*(minimal_polynomial(p))`
    # if n > 1 it contains the minimal polynomial as a factor.
    if n == 1:
        p1 = Poly(p)
        if p.coeff(x**p1.degree(x)) < 0:
            p = -p
        p = p.primitive()[1]
        return p
    # by construction `p` has root `pn`
    # the minimal polynomial is the factor vanishing in x = pn
    factors = factor_list(p)[1]

    result = _choose_factor(factors, x, pn)
    return result

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.minpoly 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] https://en.wikipedia.org/wiki/Resultant
    .. [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
           "Degrees of sums in a separable field extension".

    """
    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})
        if dom == QQ:
            R, X = ring('X', QQ)
            p1 = R(dict_from_expr(mp1)[0])
            p2 = R(dict_from_expr(mp2)[0])
        else:
            (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')

    if op is Mul or dom != QQ:
        r = resultant(mp1a, mp2, gens=[y, x])
    else:
        r = rs_compose_add(p1, p2)
        r = expr_from_dict(r.as_expr_dict(), x)

    deg1 = degree(mp1, x)
    deg2 = degree(mp2, y)
    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 _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 _minpoly_pow(ex, pw, x, dom, mp=None):
    """
    Returns ``minpoly(ex**pw, x)``

    Parameters
    ==========

    ex : algebraic element
    pw : rational number
    x : indeterminate of the polynomial
    dom: ground domain
    mp : minimal polynomial of ``p``

    Examples
    ========

    >>> from sympy import sqrt, QQ, Rational
    >>> from sympy.polys.numberfields.minpoly import _minpoly_pow, minpoly
    >>> from sympy.abc import x, y
    >>> p = sqrt(1 + sqrt(2))
    >>> _minpoly_pow(p, 2, x, QQ)
    x**2 - 2*x - 1
    >>> minpoly(p**2, x)
    x**2 - 2*x - 1
    >>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y))
    x**3 - y
    >>> minpoly(y**Rational(1, 3), x)
    x**3 - y

    """
    pw = sympify(pw)
    if not mp:
        mp = _minpoly_compose(ex, x, dom)
    if not pw.is_rational:
        raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
    if pw < 0:
        if mp == x:
            raise ZeroDivisionError('%s is zero' % ex)
        mp = _invertx(mp, x)
        if pw == -1:
            return mp
        pw = -pw
        ex = 1/ex

    y = Dummy(str(x))
    mp = mp.subs({x: y})
    n, d = pw.as_numer_denom()
    res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom)
    _, factors = res.factor_list()
    res = _choose_factor(factors, x, ex**pw, dom)
    return res.as_expr()


def _minpoly_add(x, dom, *a):
    """
    returns ``minpoly(Add(*a), dom, x)``
    """
    mp = _minpoly_op_algebraic_element(Add, a[0], a[1], x, dom)
    p = a[0] + a[1]
    for px in a[2:]:
        mp = _minpoly_op_algebraic_element(Add, p, px, x, dom, mp1=mp)
        p = p + px
    return mp


def _minpoly_mul(x, dom, *a):
    """
    returns ``minpoly(Mul(*a), dom, x)``
    """
    mp = _minpoly_op_algebraic_element(Mul, a[0], a[1], x, dom)
    p = a[0] * a[1]
    for px in a[2:]:
        mp = _minpoly_op_algebraic_element(Mul, p, px, x, dom, mp1=mp)
        p = p * px
    return mp


def _minpoly_sin(ex, x):
    """
    Returns the minimal polynomial of ``sin(ex)``
    see https://mathworld.wolfram.com/TrigonometryAngles.html
    """
    c, a = ex.args[0].as_coeff_Mul()
    if a is pi:
        if c.is_rational:
            n = c.q
            q = sympify(n)
            if q.is_prime:
                # for a = pi*p/q with q odd prime, using chebyshevt
                # write sin(q*a) = mp(sin(a))*sin(a);
                # the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
                a = dup_chebyshevt(n, ZZ)
                return Add(*[x**(n - i - 1)*a[i] for i in range(n)])
            if c.p == 1:
                if q == 9:
                    return 64*x**6 - 96*x**4 + 36*x**2 - 3

            if n % 2 == 1:
                # for a = pi*p/q with q odd, use
                # sin(q*a) = 0 to see that the minimal polynomial must be
                # a factor of dup_chebyshevt(n, ZZ)
                a = dup_chebyshevt(n, ZZ)
                a = [x**(n - i)*a[i] for i in range(n + 1)]
                r = Add(*a)
                _, factors = factor_list(r)
                res = _choose_factor(factors, x, ex)
                return res

            expr = ((1 - cos(2*c*pi))/2)**S.Half
            res = _minpoly_compose(expr, x, QQ)
            return res

    raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)


def _minpoly_cos(ex, x):
    """
    Returns the minimal polynomial of ``cos(ex)``
    see https://mathworld.wolfram.com/TrigonometryAngles.html
    """
    c, a = ex.args[0].as_coeff_Mul()
    if a is pi:
        if c.is_rational:
            if c.p == 1:
                if c.q == 7:
                    return 8*x**3 - 4*x**2 - 4*x + 1
                if c.q == 9:
                    return 8*x**3 - 6*x - 1
            elif c.p == 2:
                q = sympify(c.q)
                if q.is_prime:
                    s = _minpoly_sin(ex, x)
                    return _mexpand(s.subs({x:sqrt((1 - x)/2)}))

            # for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
            n = int(c.q)
            a = dup_chebyshevt(n, ZZ)
            a = [x**(n - i)*a[i] for i in range(n + 1)]
            r = Add(*a) - (-1)**c.p
            _, factors = factor_list(r)
            res = _choose_factor(factors, x, ex)
            return res

    raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)


def _minpoly_tan(ex, x):
    """
    Returns the minimal polynomial of ``tan(ex)``
    see https://github.com/sympy/sympy/issues/21430
    """
    c, a = ex.args[0].as_coeff_Mul()
    if a is pi:
        if c.is_rational:
            c = c * 2
            n = int(c.q)
            a = n if c.p % 2 == 0 else 1
            terms = []
            for k in range((c.p+1)%2, n+1, 2):
                terms.append(a*x**k)
                a = -(a*(n-k-1)*(n-k)) // ((k+1)*(k+2))

            r = Add(*terms)
            _, factors = factor_list(r)
            res = _choose_factor(factors, x, ex)
            return res

    raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)


def _minpoly_exp(ex, x):
    """
    Returns the minimal polynomial of ``exp(ex)``
    """
    c, a = ex.args[0].as_coeff_Mul()
    if a == I*pi:
        if c.is_rational:
            q = sympify(c.q)
            if c.p == 1 or c.p == -1:
                if q == 3:
                    return x**2 - x + 1
                if q == 4:
                    return x**4 + 1
                if q == 6:
                    return x**4 - x**2 + 1
                if q == 8:
                    return x**8 + 1
                if q == 9:
                    return x**6 - x**3 + 1
                if q == 10:
                    return x**8 - x**6 + x**4 - x**2 + 1
                if q.is_prime:
                    s = 0
                    for i in range(q):
                        s += (-x)**i
                    return s

            # x**(2*q) = product(factors)
            factors = [cyclotomic_poly(i, x) for i in divisors(2*q)]
            mp = _choose_factor(factors, x, ex)
            return mp
        else:
            raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
    raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)


def _minpoly_rootof(ex, x):
    """
    Returns the minimal polynomial of a ``CRootOf`` object.
    """
    p = ex.expr
    p = p.subs({ex.poly.gens[0]:x})
    _, factors = factor_list(p, x)
    result = _choose_factor(factors, x, ex)
    return result


def _minpoly_compose(ex, x, dom):
    """
    Computes the minimal polynomial of an algebraic element
    using operations on minimal polynomials

    Examples
    ========

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

    """
    if ex.is_Rational:
        return ex.q*x - ex.p
    if ex is I:
        _, factors = factor_list(x**2 + 1, x, domain=dom)
        return x**2 + 1 if len(factors) == 1 else x - I

    if ex is S.GoldenRatio:
        _, factors = factor_list(x**2 - x - 1, x, domain=dom)
        if len(factors) == 1:
            return x**2 - x - 1
        else:
            return _choose_factor(factors, x, (1 + sqrt(5))/2, dom=dom)

    if ex is S.TribonacciConstant:
        _, factors = factor_list(x**3 - x**2 - x - 1, x, domain=dom)
        if len(factors) == 1:
            return x**3 - x**2 - x - 1
        else:
            fac = (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
            return _choose_factor(factors, x, fac, dom=dom)

    if hasattr(dom, 'symbols') and ex in dom.symbols:
        return x - ex

    if dom.is_QQ and _is_sum_surds(ex):
        # eliminate the square roots
        ex -= x
        while 1:
            ex1 = _separate_sq(ex)
            if ex1 is ex:
                return ex
            else:
                ex = ex1

    if ex.is_Add:
        res = _minpoly_add(x, dom, *ex.args)
    elif ex.is_Mul:
        f = Factors(ex).factors
        r = sift(f.items(), lambda itx: itx[0].is_Rational and itx[1].is_Rational)
        if r[True] and dom == QQ:
            ex1 = Mul(*[bx**ex for bx, ex in r[False] + r[None]])
            r1 = dict(r[True])
            dens = [y.q for y in r1.values()]
            lcmdens = reduce(lcm, dens, 1)
            neg1 = S.NegativeOne
            expn1 = r1.pop(neg1, S.Zero)
            nums = [base**(y.p*lcmdens // y.q) for base, y in r1.items()]
            ex2 = Mul(*nums)
            mp1 = minimal_polynomial(ex1, x)
            # use the fact that in SymPy canonicalization products of integers
            # raised to rational powers are organized in relatively prime
            # bases, and that in ``base**(n/d)`` a perfect power is
            # simplified with the root
            # Powers of -1 have to be treated separately to preserve sign.
            mp2 = ex2.q*x**lcmdens - ex2.p*neg1**(expn1*lcmdens)
            ex2 = neg1**expn1 * ex2**Rational(1, lcmdens)
            res = _minpoly_op_algebraic_element(Mul, ex1, ex2, x, dom, mp1=mp1, mp2=mp2)
        else:
            res = _minpoly_mul(x, dom, *ex.args)
    elif ex.is_Pow:
        res = _minpoly_pow(ex.base, ex.exp, x, dom)
    elif ex.__class__ is sin:
        res = _minpoly_sin(ex, x)
    elif ex.__class__ is cos:
        res = _minpoly_cos(ex, x)
    elif ex.__class__ is tan:
        res = _minpoly_tan(ex, x)
    elif ex.__class__ is exp:
        res = _minpoly_exp(ex, x)
    elif ex.__class__ is CRootOf:
        res = _minpoly_rootof(ex, x)
    else:
        raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
    return res


@public
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

    """

    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 _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

    """

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

    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):
        """
        Transform a given algebraic expression *ex* into a multivariate
        polynomial, by introducing fresh variables with defining equations.

        Explanation
        ===========

        The critical elements of the algebraic expression *ex* are root
        extractions, instances of :py:class:`~.AlgebraicNumber`, and negative
        powers.

        When we encounter a root extraction or an :py:class:`~.AlgebraicNumber`
        we replace this expression with a fresh variable ``a_i``, and record
        the defining polynomial for ``a_i``. For example, if ``a_0**(1/3)``
        occurs, we will replace it with ``a_1``, and record the new defining
        polynomial ``a_1**3 - a_0``.

        When we encounter a negative power we transform it into a positive
        power by algebraically inverting the base. This means computing the
        minimal polynomial in ``x`` for the base, inverting ``x`` modulo this
        poly (which generates a new polynomial) and then substituting the
        original base expression for ``x`` in this last polynomial.

        We return the transformed expression, and we record the defining
        equations for new symbols using the ``update_mapping()`` function.

        """
        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:
                    minpoly_base = _minpoly_groebner(ex.base, x, cls)
                    inverse = invert(x, minpoly_base).as_expr()
                    base_inv = inverse.subs(x, ex.base).expand()

                    if ex.exp == -1:
                        return bottom_up_scan(base_inv)
                    else:
                        ex = base_inv**(-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:
                    if exp.is_Integer:
                        return expr.expand()
                    else:
                        return update_mapping(expr, 1 / exp, -base)
                else:
                    return symbols[expr]
        elif ex.is_AlgebraicNumber:
            if ex not in mapping:
                return update_mapping(ex, ex.minpoly_of_element())
            else:
                return symbols[ex]

        raise NotAlgebraic("%s does not 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
            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:
                return True
        return False

    inverted = False
    ex = expand_multinomial(ex)
    if ex.is_AlgebraicNumber:
        return ex.minpoly_of_element().as_expr(x)
    elif ex.is_Rational:
        result = ex.q*x - ex.p
    else:
        inverted = simpler_inverse(ex)
        if inverted:
            ex = ex**-1
        res = None
        if ex.is_Pow and (1/ex.exp).is_Integer:
            n = 1/ex.exp
            res = _minimal_polynomial_sq(ex.base, n, x)

        elif _is_sum_surds(ex):
            res = _minimal_polynomial_sq(ex, S.One, x)

        if res is not None:
            result = res

        if res is None:
            bus = bottom_up_scan(ex)
            F = [x - bus] + list(mapping.values())
            G = groebner(F, list(symbols.values()) + [x], order='lex')

            _, factors = factor_list(G[-1])
            # by construction G[-1] has root `ex`
            result = _choose_factor(factors, x, ex)
    if inverted:
        result = _invertx(result, x)
        if result.coeff(x**degree(result, x)) < 0:
            result = expand_mul(-result)

    return result


@public
def minpoly(ex, x=None, compose=True, polys=False, domain=None):
    """This is a synonym for :py:func:`~.minimal_polynomial`."""
    return minimal_polynomial(ex, x=x, compose=compose, polys=polys, domain=domain)