It’s a bit more involved than that; see the parallel discussion of the same
problem in this post
<https://groups.google.com/g/sage-support/c/ZnKV3D-i9t0> on Sagemath
support list.
Le lundi 6 décembre 2021 à 02:28:31 UTC+1, [email protected] a écrit :
> It just looks like the OP issue has two extra solutions (when the initial
> equations are not factored) which couldn't be eliminated (easily) and the
> solver sends them back to be safe.
>
> NOTE: `fsol` in the previous was obtained with `solve([factor(i) for i in
> Sys], Unk, simplify=False)`.
>
> If checking is turned off then two invalid solutions are included, but the
> solution involving `sqrt(303)` is missing:
> ```python
> >>> s=solve([factor(i) for i in Sys], Unk, check=0,dict=1); s
> [{a2: 0, a1: 0}, {a2: 0, a1: 0}, {a1: 0, a2: 0}, {a1: 0, b2: 0}, {a2: 0,
> b2: 0}, {a2: a1**2, b1: b2*(a1**3 + 2*a1**2 - 1)/(3*a1**3)}]
> >>> [[e.subs(si).simplify() for e in Sys] for si in s]
> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, a2**3, -2*a2**2*b1], [0,
> 0, 0, 0], [0, 0, 0, a1*b2*(a1**3 - 4*a1**2 + 2)/3]]
> ```
>
> /c
> On Sunday, December 5, 2021 at 8:50:27 AM UTC-6 Chris Smith wrote:
>
>> If you factor the equations before passing them to solve then the
>> solution is
>> ```
>> >>> fsol
>> [{a2: 0, a1: 0}, {a2: 0, a1: 0}, {a2: 0, b2: 0}, {a2: 0, b1: b2/2, a1:
>> 0}, {a2: (64 + (-8 + (1 - sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(1 -
>> sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))**2/(36*(1 - sqrt(3)*I)**2*(37
>> + 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: (-64 + (1 - sqrt(3)*I)*(8 +
>> (-1 + sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(37 +
>> 3*sqrt(303)*I)**(1/3))/(6*(1 - sqrt(3)*I)*(37 +
>> 3*sqrt(303)*I)**(1/3))}, {a2: (64 + (-8 + (1 + sqrt(3)*I)*(37 +
>> 3*sqrt(303)*I)**(1/3))*(1 + sqrt(3)*I)*(37 +
>> 3*sqrt(303)*I)**(1/3))**2/(36*(1 + sqrt(3)*I)**2*(37 +
>> 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: (-64 + (1 + sqrt(3)*I)*(8 - (1 +
>> sqrt(3)*I)*(37 + 3*sqrt(303)*I)**(1/3))*(37 +
>> 3*sqrt(303)*I)**(1/3))/(6*(1 + sqrt(3)*I)*(37 +
>> 3*sqrt(303)*I)**(1/3))}, {a2: (16 + (4 + (37 +
>> 3*sqrt(303)*I)**(1/3))*(37 + 3*sqrt(303)*I)**(1/3))**2/(9*(37 +
>> 3*sqrt(303)*I)**(2/3)), b1: b2/2, a1: 4/3 + 16/(3*(37 +
>> 3*sqrt(303)*I)**(1/3)) + (37 + 3*sqrt(303)*I)**(1/3)/3}, {a2: a1**2,
>> b1: 0, b2: 0}]
>> ```
>> and all of them are true solutions:
>> ```
>> >>> [[e.subs(s).simplify() for e in Sys] for s in fsol]
>> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0],
>> [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]
>> ```
>>
>> On Sunday, December 5, 2021 at 8:35:28 AM UTC-6 Chris Smith wrote:
>>
>>> `nonlinsolve(Sys, Unk)` gives
>>> ```
>>> {(0, 0, b1, b2), (0, 0, b2/2, b2), (a1, 0, b1, 0), (-sqrt(a2), a2, 0,
>>> 0), (sqrt(a2), a2, 0, 0)}
>>> ```
>>> On Saturday, December 4, 2021 at 4:36:04 PM UTC-6 [email protected]
>>> wrote:
>>>
>>>> A bug in Sympy’s solve
>>>>
>>>> Inspired by this ask.sagemath.orgquestion
>>>> <https://ask.sagemath.org/question/59063/weird-c-values-from-solving-system-of-equations/>
>>>> .
>>>>
>>>> This question was about the names of the symbolic variables created by
>>>> Maxima’s solve to denote arbitrary constants. Exploring the example
>>>> used in this questions revealed a non-trivial problem with sympy’s
>>>> solve.
>>>> Problem
>>>>
>>>> Solve the system :
>>>> $$
>>>> \begin{align*}
>>>>
>>>> - a*{1}^{3} a*{2} + a*{1} a*{2}^{2} &= 0 \
>>>> - 3 a*{1}^{2} a*{2} b*{1} + 2 a*{1} a*{2} b*{2} - a*{1} b*{2} +
>>>> a*{2}^{2}
>>>> b*{2} &= 0 \
>>>> - a*{1}^{2} a*{2}^{2} + a_{2}^{3} &= 0 \
>>>> - 2 a*{1}^{2} a*{2} b*{2} - 2 a*{2}^{2} b*{1} + 3 a*{2}^{2} b_{2}
>>>> &= 0
>>>> \end{align*}
>>>> $$
>>>>
>>>> # Set up sympy (brutal version)import sympyfrom sympy import *
>>>> init_session()
>>>> init_printing(pretty_print=False)# System to solve
>>>> a1, a2, b1, b2 = symbols('a1 a2 b1 b2')
>>>> Unk = [a1, a2, b1, b2]
>>>> eq1 = a1 * a2**2 - a2 * a1**3
>>>> eq2 = 2*a1*a2*b2 + b2*a2**2 - 3*a2*a1**2*b1 - a1*b2
>>>> eq3 = a2**3 - a2**2*a1**2
>>>> eq4 = 3*a2**2*b2 - 2*a2*a1**2*b2 - 2*a2**2*b1
>>>> Sys = [eq1, eq2, eq3, eq4]
>>>>
>>>> IPython console for SymPy 1.9 (Python 3.9.9-64-bit) (ground types: gmpy)
>>>>
>>>> These commands were executed:
>>>> >>> from __future__ import division
>>>> >>> from sympy import *
>>>> >>> x, y, z, t = symbols('x y z t')
>>>> >>> k, m, n = symbols('k m n', integer=True)
>>>> >>> f, g, h = symbols('f g h', cls=Function)
>>>> >>> init_printing()
>>>>
>>>> Documentation can be found at https://docs.sympy.org/1.9/
>>>>
>>>> Attempt to use the “automatic” Sympy solver:
>>>>
>>>> Sol = solve(Sys, Unk)
>>>> DSol = [dict(zip(Unk, u)) for u in Sol]
>>>> DSol
>>>>
>>>> [{a1: 0, a2: 0, b1: b1, b2: b2}, {a1: 0, a2: 0, b1: b1, b2: b2}, {a1: 0,
>>>> a2: 0, b1: b1, b2: b2}, {a1: 0, a2: 0, b1: b2/2, b2: b2}, {a1: a1, a2: 0,
>>>> b1: b1, b2: 0}, {a1: -sqrt(a2), a2: a2, b1: 0, b2: 0}, {a1:
>>>> -2**(5/6)*sqrt(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3) +
>>>> 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6, a2: 16/3 +
>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3, b1: b2/2, b2: b2}, {a1: -sqrt(6)*sqrt(32 -
>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1
>>>> + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/6, a2: 2**(2/3)*(-208/3 + (1
>>>> + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)/((1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)),
>>>> b1: b2/2, b2: b2}, {a1: -sqrt(6)*sqrt(32 - 416*2**(2/3)/((1 -
>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499
>>>> + 3*sqrt(303)*I)**(1/3))/6, a2: 2**(2/3)*(-208/3 + (1 - sqrt(3)*I)*(32 +
>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)/((1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)),
>>>> b1: b2/2, b2: b2}]
>>>>
>>>> Something went sideways: the six first solutions are okay,but the last
>>>> three use expressions, some of them being polynomials in b2.
>>>>
>>>> Attempt to check them formally :
>>>>
>>>> Chk=[[u.subs(s) for u in Sys] for s in DSol]
>>>> Chk
>>>>
>>>> [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0,
>>>> 0, 0, 0], [-2**(5/6)*(16/3 + 104*2**(2/3)/(3*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)**2*sqrt(48*2**(1/3) + 624/(1499 +
>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6 +
>>>> sqrt(2)*(16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) +
>>>> 2**(1/3)*(1499 + 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 +
>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3))**(3/2)/54, -2**(2/3)*b2*(16/3 +
>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3)
>>>> + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/12 - 2**(5/6)*b2*(16/3 +
>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)*sqrt(48*2**(1/3) + 624/(1499 +
>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/3 +
>>>> b2*(16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499
>>>> + 3*sqrt(303)*I)**(1/3)/3)**2 + 2**(5/6)*b2*sqrt(48*2**(1/3) + 624/(1499 +
>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/6,
>>>> -2**(2/3)*(16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) +
>>>> 2**(1/3)*(1499 + 3*sqrt(303)*I)**(1/3)/3)**2*(48*2**(1/3) + 624/(1499 +
>>>> 3*sqrt(303)*I)**(1/3) + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/18 +
>>>> (16/3 + 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)**3, -2**(2/3)*b2*(16/3 + 104*2**(2/3)/(3*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)*(48*2**(1/3) + 624/(1499 + 3*sqrt(303)*I)**(1/3)
>>>> + 3*2**(2/3)*(1499 + 3*sqrt(303)*I)**(1/3))/9 + 2*b2*(16/3 +
>>>> 104*2**(2/3)/(3*(1499 + 3*sqrt(303)*I)**(1/3)) + 2**(1/3)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/3)**2], [2**(1/6)*sqrt(3)*(-208/3 + (1 +
>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)))**(3/2)/(18*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) - 2**(5/6)*sqrt(3)*(-208/3 + (1 + sqrt(3)*I)*(32 +
>>>> (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)**2*sqrt(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)))/(3*(1 + sqrt(3)*I)**2*(1499 +
>>>> 3*sqrt(303)*I)**(2/3)), -2*2**(1/6)*sqrt(3)*b2*(-208/3 + (1 +
>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*sqrt(32 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)))/(3*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))
>>>> - 3*2**(2/3)*b2*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)*(16/3 -
>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)/6 -
>>>> 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/(2*(1 +
>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) + sqrt(6)*b2*sqrt(32 -
>>>> 2**(1/3)*(1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3) - 416*2**(2/3)/((1
>>>> + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)))/6 + 2*2**(1/3)*b2*(-208/3 +
>>>> (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2/((1 +
>>>> sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3)), -2*2**(1/3)*(-208/3 + (1 +
>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)**2*(16/3 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/6 - 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)))/((1 + sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3))
>>>> + 4*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**3/((1 +
>>>> sqrt(3)*I)**3*(1499 + 3*sqrt(303)*I)), -2*2**(2/3)*b2*(-208/3 + (1 +
>>>> sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*(16/3 - 2**(1/3)*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/6 - 208*2**(2/3)/(3*(1 + sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)))/((1 + sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) +
>>>> 4*2**(1/3)*b2*(-208/3 + (1 + sqrt(3)*I)*(32 + (-1 - sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2/((1 +
>>>> sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3))], [-2**(5/6)*sqrt(3)*(-208/3
>>>> + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2*sqrt(32 -
>>>> 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1
>>>> - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))/(3*(1 - sqrt(3)*I)**2*(1499 +
>>>> 3*sqrt(303)*I)**(2/3)) + 2**(1/6)*sqrt(3)*(-208/3 + (1 - sqrt(3)*I)*(32 +
>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*(32 - 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3))**(3/2)/(18*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)), sqrt(6)*b2*sqrt(32 - 416*2**(2/3)/((1 -
>>>> sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499
>>>> + 3*sqrt(303)*I)**(1/3))/6 + 2*2**(1/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 +
>>>> (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)**2/((1 - sqrt(3)*I)**2*(1499 +
>>>> 3*sqrt(303)*I)**(2/3)) - 3*2**(2/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1
>>>> + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*(16/3 - 208*2**(2/3)/(3*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/6)/(2*(1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3))
>>>> - 2*2**(1/6)*sqrt(3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 +
>>>> sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*sqrt(32 - 416*2**(2/3)/((1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3))/(3*(1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)),
>>>> 4*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**3/((1 -
>>>> sqrt(3)*I)**3*(1499 + 3*sqrt(303)*I)) - 2*2**(1/3)*(-208/3 + (1 -
>>>> sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)**2*(16/3 - 208*2**(2/3)/(3*(1 - sqrt(3)*I)*(1499
>>>> + 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/6)/((1 - sqrt(3)*I)**2*(1499 +
>>>> 3*sqrt(303)*I)**(2/3)), -2*2**(2/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1
>>>> + sqrt(3)*I)*(2998 + 6*sqrt(303)*I)**(1/3))*(2998 +
>>>> 6*sqrt(303)*I)**(1/3)/12)*(16/3 - 208*2**(2/3)/(3*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)) - 2**(1/3)*(1 - sqrt(3)*I)*(1499 +
>>>> 3*sqrt(303)*I)**(1/3)/6)/((1 - sqrt(3)*I)*(1499 + 3*sqrt(303)*I)**(1/3)) +
>>>> 4*2**(1/3)*b2*(-208/3 + (1 - sqrt(3)*I)*(32 + (-1 + sqrt(3)*I)*(2998 +
>>>> 6*sqrt(303)*I)**(1/3))*(2998 + 6*sqrt(303)*I)**(1/3)/12)**2/((1 -
>>>> sqrt(3)*I)**2*(1499 + 3*sqrt(303)*I)**(2/3))]]
>>>>
>>>> Some of these expressions are polynomial in b2 whose coefficients of
>>>> not-null dregree are nort *obviously* null.
>>>> A bit of hit-and-miss trials leads to this attempt at *numerical*
>>>> check :
>>>>
>>>> def chz(x):
>>>> r = x.factor().is_zero
>>>> if r is None: return x.coeff(b2).n()
>>>> return r
>>>> [[chz(u) for u in v] for v in Chk]
>>>>
>>>> [[True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, -223.427427555427 + 0.e-25*I, True, True],
>>>> [True, -0.388012232966408 - 5.25038392589403e-29*I, True, True],
>>>> [True, -0.e-137 + 1.12445821450677e-139*I, True, True]]
>>>>
>>>> Solution #8 may be exactr, but #6 and #7 cannot.
>>>> Manual solution.
>>>>
>>>> eq1 and eq3 give us solutions for a1 and a2 :
>>>>
>>>> S13=solve([eq3, eq1], [a2, a1], dict=True) ; S13
>>>>
>>>> [{a1: -sqrt(a2)}, {a1: sqrt(a2)}, {a2: 0}]
>>>>
>>>> which we prefer to rewrite as :
>>>>
>>>> S13 = [{a2:a1**2}, {a2:0}]; S13
>>>>
>>>> [{a2: a1**2}, {a2: 0}]
>>>>
>>>> Substituting these values in eq4 gives us :
>>>>
>>>> E4 = [eq4.subs(s) for s in S13] ; E4
>>>>
>>>> [-2*a1**4*b1 + a1**4*b2, 0]
>>>>
>>>> The second solution tells us that S13[1] is also,a solution to [eq1,
>>>> eq3, eq4].
>>>>
>>>> S134=S13[1:] ; S134
>>>>
>>>> [{a2: 0}]
>>>>
>>>> Substituting S13[0] ineq4and solving for the variables of the
>>>> resulting polynomial augments the setS134of solutions of[eq1, eq3`,
>>>> eq4]’ :
>>>>
>>>> V4=E4[0].free_symbols
>>>> S4=[solve(E4[0], v, dict=True) for v in V4] ; S4for S in S4:
>>>> for s in S:
>>>> S0={u:S13[0][u].subs(s) for u in S13[0].keys()}
>>>> S134 += [S0.copy()|s]
>>>> S134
>>>>
>>>> [{a2: 0}, {a2: a1**2, b1: b2/2}, {a1: 0, a2: 0}, {a2: a1**2, b2: 2*b1}]
>>>>
>>>> Again, we prefer to rewrite it in a simpler (and shorter) fashion :
>>>>
>>>> S134=[{a2:0}, {a2:a1**2, b2:2*b1}] ; S134
>>>>
>>>> [{a2: 0}, {a2: a1**2, b2: 2*b1}]
>>>>
>>>> Substituting in eq3 gives :
>>>>
>>>> [eq2.subs(s) for s in S134]
>>>>
>>>> [-a1*b2, -a1**4*b1 + 4*a1**3*b1 - 2*a1*b1]
>>>>
>>>> Solving these equations for their free symbols a,d merging with the
>>>> previous partial solutions gives us the solutions of the full system :
>>>>
>>>> S1234=[]for S in S134:
>>>> # print("S=",S)
>>>> E=eq2.subs(S)
>>>> # print("E=",E)
>>>> S1=[solve(E, v, dict=True) for v in E.free_symbols]
>>>> # print("S1=",S1)
>>>> for s in flatten(S1):
>>>> # print("s=",s)
>>>> S0={u:S[u].subs(s) if "subs" in dir(S[u]) else S[u] for u in
>>>> S.keys()}
>>>> S1234+=[S0.copy()|s]
>>>> S1234
>>>>
>>>> [{a1: 0, a2: 0}, {a2: 0, b2: 0}, {a2: a1**2, b1: 0, b2: 0}, {a1: 0, a2: 0,
>>>> b2: 2*b1}, {a1: 4/3 + (-1/2 - sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)
>>>> + 16/(9*(-1/2 - sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)), a2: (4/3 +
>>>> (-1/2 - sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3) + 16/(9*(-1/2 -
>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)))**2, b2: 2*b1}, {a1: 4/3 +
>>>> 16/(9*(-1/2 + sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)) + (-1/2 +
>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3), a2: (4/3 + 16/(9*(-1/2 +
>>>> sqrt(3)*I/2)*(37/27 + sqrt(303)*I/9)**(1/3)) + (-1/2 + sqrt(3)*I/2)*(37/27
>>>> + sqrt(303)*I/9)**(1/3))**2, b2: 2*b1}, {a1: 4/3 + 16/(9*(37/27 +
>>>> sqrt(303)*I/9)**(1/3)) + (37/27 + sqrt(303)*I/9)**(1/3), a2: (4/3 +
>>>> 16/(9*(37/27 + sqrt(303)*I/9)**(1/3)) + (37/27 +
>>>> sqrt(303)*I/9)**(1/3))**2, b2: 2*b1}]
>>>>
>>>> Again, some solutions, substituted in eq2, give first-degree monomials
>>>> in b1 whose oefficient cannot be shownt to be null byis_zero :
>>>>
>>>> [[e.subs(s).is_zero for e in Sys] for s in S1234]
>>>>
>>>> [[True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, True, True, True],
>>>> [True, None, True, True],
>>>> [True, None, True, True],
>>>> [True, None, True, True]]
>>>>
>>>> But, this time, the numerical check points to a probably null result :
>>>>
>>>> [Sys[1].subs(S1234[u]).coeff(b1).n() for u in range(3,6)]
>>>>
>>>> [0, 0.e-125 + 0.e-127*I, 0.e-125 - 0.e-127*I]
>>>>
>>>>
>>>>
>>>
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