Solve the system $x^3+y=3x+4$, $2y^3+z=6y+6$, $3z^3+x=9z+8$

Question

Find the real solutions of the system below:

$$\begin{aligned} x^{3}+y &= 3x+4\\ 2y^{3}+z &= 6y+6\\ 3z^{3}+x &= 9z+8\end{aligned}$$

I wrote the system as: $$x(x^{2}-3)=4-y$$ $$2y(y^{2}-3)=6-z$$ $$3z(z^{2}-3)=8-x$$ Then I tried to use the addition and multiplication of the equations above, but it didn't seem to be very useful.

Answer 1

We have $$(y-2)=-(x-2)(x+1)^2\,,$$ $$(z-2)=-2(y-2)(y+1)^2\,,$$ and $$(x-2)=-3(z-2)(z+1)^2\,.$$ Multiply the three equation above to get $$(x-2)(y-2)(z-2)\big(1+6(x+1)^2(y+1)^2(z+1)^2\big)=0\,.$$ Since $1+6(x+1)^2(y+1)^2(z+1)^2\geq 1>0$, we conclude that $$(x-2)(y-2)(z-2)=0\,,$$ whence $x=2$, $y=2$, or $z=2$. However, this means $x=y=z=2$.

Answer 2

Using SymPy:

>>> from sympy import *
>>> x, y, z = symbols('x y z', real=True)
>>> p1 = x**3 + y - 3*x - 4
>>> p2 = 2*y**3 + z - 6*y - 6
>>> p3 = 3*z**3 + x - 9*z - 8
>>> solve_poly_system([p1,p2,p3],x,y,z)
[(2, 2, 2)]

Computing a Gröbner basis using Buchberger’s algorithm:

>>> groebner([p1,p2,p3],x,y,z,order='lex')
GroebnerBasis([x + 3*z**3 - 9*z - 8, y - 27*z**9 + 243*z**7 + 216*z**6 - 729*z**5 - 1296*z**4 + 162*z**3 + 1944*z**2 + 1701*z + 484, 39366*z**27 - 1062882*z**25 - 944784*z**24 + 12754584*z**23 + 22674816*z**22 - 79243758*z**21 - 238085568*z**20 + 190964466*z**19 + 1366560072*z**18 + 691936182*z**17 - 4241765232*z**16 - 6830928288*z**15 + 4492920312*z**14 + 21689143848*z**13 + 13528816992*z**12 - 26880128316*z**11 - 51050721096*z**10 - 14498247996*z**9 + 49512350304*z**8 + 67115915712*z**7 + 22832591040*z**6 - 29383005474*z**5 - 44502137424*z**4 - 29276615214*z**3 - 11134786824*z**2 - 2390806529*z - 226756910], x, y, z, domain='ZZ', order='lex')

Note that we have a polynomial in $z$ of degree $27$. SymPy found only a single root, namely, $z=2$.