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### Solving a polynomial system in a quotient ring

I want to compute all solutions over $\mathbb{Z}_9$ in parameters $a,b,c$ for the equation

$$(ax+b\sqrt2+c)^2=1$$

where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$.

I'm first defining the polynomial ring over $\mathbb{Z}_9$ in variables $a,b,c,x,y$, then factoring by the ideal generated by

$$y^2-2, (x+y)^2-2(x+y),$$

but then I don't know which command to use in order to get the solutions. I have tried "solve" and "variety", but they do not seem to work. The code up to this point is just

R.<x,y,a,b,c> = PolynomialRing(IntegerModRing(9),order='lex')
J= R.ideal(x^2-2,(x+y)^2-2*(x+y))
S=R.quotient(J)


Which function should I use?

### Solving a polynomial system in a quotient ring

I want to compute all solutions over $\mathbb{Z}_9$ in parameters $a,b,c$ for of the equation

$$(ax+b\sqrt2+c)^2=1$$

where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$.

I'm first defining the polynomial ring over $\mathbb{Z}_9$ in variables $a,b,c,x,y$, then factoring by the ideal generated by

$$y^2-2, (x+y)^2-2(x+y),$$

but then I don't know which command to use in order to get the solutions. I have tried "solve" and "variety", but they do not seem to work. The code up to this point is just

R.<x,y,a,b,c> = PolynomialRing(IntegerModRing(9),order='lex')
J= R.ideal(x^2-2,(x+y)^2-2*(x+y))
S=R.quotient(J)


Which function should I use?

### Solving a polynomial system in a quotient ring

I want to compute all solutions over $\mathbb{Z}_9$ in parameters $a,b,c$ of the equation

$$(ax+b\sqrt2+c)^2=1$$

where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$.

I'm first defining the polynomial ring over $\mathbb{Z}_9$ in variables $a,b,c,x,y$, then factoring by the ideal generated by

$$y^2-2, (x+y)^2-2(x+y),$$

but then I don't know which command to use in order to get the solutions. I have tried "solve" and "variety", but they do not seem to work. The code up to this point is just

R.<x,y,a,b,c> = PolynomialRing(IntegerModRing(9),order='lex')
J= R.ideal(x^2-2,(x+y)^2-2*(x+y))
S=R.quotient(J)


Which function should I use?

### Solving a polynomial system in a quotient ring

I want to compute all solutions over $\mathbb{Z}_9$ in parameters $a,b,c$ $\mathbb{Z}_9[\sqrt2,x]$, where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$, of the equation

$$(ax+b\sqrt2+c)^2=1$$

where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$.$$X^2=1.$$

I'm first defining the polynomial ring over $\mathbb{Z}_9$ in variables $a,b,c,x,y$, $x,y$, then factoring by the ideal generated by

$$y^2-2, (x+y)^2-2(x+y),$$

but then I don't know which command to use in order to get the solutions. solutions of $X^2-1$. I have tried "solve" and "variety", but they do not seem to work. The code up to this point is just

R.<x,y,a,b,c> R.<x,y> = PolynomialRing(IntegerModRing(9),order='lex')
J= R.ideal(x^2-2,(x+y)^2-2*(x+y))
S=R.quotient(J)


Which function should I use?

### Solving a polynomial system in a quotient ring

I want to compute all solutions in $\mathbb{Z}_9[\sqrt2,x]$, where $x$ is such that $(x+\sqrt2)^2=2(x+\sqrt2)$, of the equation

$$X^2=1.$$

I'm first defining the polynomial ring over $\mathbb{Z}_9$ in variables $x,y$, then factoring by the ideal generated by

$$y^2-2, (x+y)^2-2(x+y),$$

to get the ring $S$, but then I don't know which command to use in order to get the solutions of $X^2-1$. I have tried "solve" and "variety", "variety" (defining $S[X]$ first and then the ideal of $X^2-1$), but they do not seem to work. The code up to this point is just

R.<x,y> = PolynomialRing(IntegerModRing(9),order='lex')
J= R.ideal(x^2-2,(x+y)^2-2*(x+y))
S=R.quotient(J)


Which function should I use?