Solving a polynomial system in a quotient ring

edit

Here is a possibility to work.

Let $A$ be a ring such that $2$ is invertible, for example $\Bbb Z/3$ or $\Bbb Z/9$. We denote by $\frac 12$ its inverse. Assume $2$ is not a square in $A$. (This is the case in the above examples.)

Consider $A[a]$ to be the ring $A[Y]/(Y^2-2)$, where $A[Y]$ is the polynomial ring in $Y$ over $A$. (And $a$ is $Y$ modulo the ideal generated by $(Y^2-2)$. So formally, $a=\sqrt2$.

Now observe that the equation satisfied by the $x$ element in the OP, $(x+a)^2=2(x+a)$, can be written as $$(x+a)(x+a-2)=0\ .$$ The two factors are relatively prime, in fact $$\frac 12(x+a)-\frac 12(x+a-2) = 1\ .$$ This means that the ring $$ R=A[a][X]\ /\ (\ (X+a)(X+a-2)\ ) $$ is isomorphic to two copies of $A[a]$ via the map $$ R\to A[a]\times A[a]\ ,\ f(X)\to(f(-a), f(2-a))\ . $$ The inverse map takes $(s,t)\in A[a]\times A[a]$ and maps it into $\frac 12t(X+a)-\frac 12s(X+a-2)$.

Now we have to solve $Z^2=1$ in the ring $A[a]\times A[a]$. This can be done also manually. An element of the shape $Z=u+va\in A[a]$, $u,v\in A$ satisfies $Z^2 = 1$ iff $(u^2+2v^2)+2auv=1$. In case $A$ has zero divisors we have to take care of $uv=0$ somehow. The possible values for $u,v$ that may lead to a solution satisfy at any rate $u^3=u$ and $2v^3=v$. Together with $Z$, also its "conjugate" $\bar Z=u-va$ is also a solution, and the "norm" $N(Z)=Z\bar Z=(u+va)(u-va)=u^2-2v^2$ is $1$. So it is a good idea to search for solutions of this "Pell equation" over $A$.

Let us now write some lines of code for the given case.

The brute force search is:

r = Zmod(9)
R.<a,x> = PolynomialRing(r, order='lex')
J = R.ideal( [a^2-2, (x+a)^2-2*(x+a)] )
S = R.quotient(J)

for [s, t, u] in cartesian_product([r, r, r]):
    Z = S(s + t*a + u*x)
    if Z^2 == S(1):
        print(f"Z = {s} + {t} a + {u} x")

Results:

Z = 1 + 0 a + 0 x
Z = 1 + 8 a + 8 x
Z = 8 + 0 a + 0 x
Z = 8 + 1 a + 1 x

This fits with the situation of finding all points $Z=(Z_1,Z_2)$ over the ring $R=A[a]=\Bbb Z/9[a]=\Bbb Z/9[\sqrt 2]$ with $Z^2=(1,1)=1_R$.

sage: r = Integers(9)                                                                                                         
sage: R.<Y> = r[]                                                                                                             
sage: Q.<a> = R.quotient(Y^2-2)                                                                                               
sage: a^2                                                                                                                     
2
sage: for r1, r2 in cartesian_product([r, r]): 
....:     Z1 = r1 + r2*a 
....:     if Z1^2 == Q(1): 
....:         print(Z1) 
....:                                                                                                                         
1
8

These are the only Hensel lifts of the corresponding units over $\Bbb Z/3$:

sage: U.<A> = PolynomialRing(GF(3))                                                                                           
sage: F.<a> = GF(3^2, modulus = A^2-2)                                                                                        
sage: a^2                                                                                                                     
2
sage: [ f for f in F if f^2 == F(1) ]                                                                                         
[2, 1]
sage: # of course, this is a field...