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, smi...@gmail.com 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 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] >>>> >>>> >>>> >>> -- You received this message because you are subscribed to the Google Groups "sympy" group. 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