`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 emanuel.c...@gmail.com
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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