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I'm VERY new to SageMath, and I'm having trouble understanding how polynomial rings work in this language... I have the following code:

sage: F = PolynomialRing(GF(3),'x'); x = F.gen()
sage: S = F.quotient(x^2,'a'); a = S.gen()

I'm trying to find solutions to the following matrix equation:

sage: var('d1,d2,d3')
sage: var('r1,r2,r3')
sage: laplacian = matrix([[d1, -1, 0], [-1, d2, -1], [0, -1, d3]])
sage: rstructure = laplacian.solve_right(vector([0,0,0]))

Now here is the caveat: I want the solutions of rstructure to only be elements of the quotient ring I've defined as S. If you've gotten this far, I'll boil down my main questions below:

(1) What are x=F.gen() and a=S.gen() doing?

(2) I want to see all the possible solutions of r_i, d_i that satisfy the matrix equation with the condition that they are elements of S (the quotient ring that I've defined earlier).

(3) If it is not clear, "rstructure" is an element of (F_3[x]/< x^2 >)^3. What I'm particularly interested in are solutions in (F_p[x])^3. Is computing this even possible for a given prime p?

Thanks for taking the time to read this! Hope to figure this out soon.

I'm VERY new to SageMath, and I'm having trouble understanding how polynomial rings work in this language... I have the following code:

sage: F = PolynomialRing(GF(3),'x'); x = F.gen()
sage: S = F.quotient(x^2,'a'); a = S.gen()

I'm trying to find solutions to the following matrix equation:

sage: var('d1,d2,d3')
sage: var('r1,r2,r3')
sage: laplacian = matrix([[d1, -1, 0], [-1, d2, -1], [0, -1, d3]])
sage: rstructure = laplacian.solve_right(vector([0,0,0]))

Now here is the caveat: I want the solutions of rstructure to only be elements of the quotient ring I've defined as S. If you've gotten this far, I'll boil down my main questions below:

(1) What are x=F.gen() and a=S.gen() doing?

(2) I want to see all the possible solutions of r_i, d_i that satisfy the matrix equation with the condition that they are elements of S (the quotient ring that I've defined earlier).

(3) If it is not clear, "rstructure" is an element of (F_3[x]/< x^2 >)^3. What I'm particularly interested in are solutions in (F_p[x])^3. Is computing this even possible for a given prime p?

Thanks for taking the time to read this! Hope to figure this out soon.

I'm VERY new to SageMath, and I'm having trouble understanding how polynomial rings work in this language... I have the following code:

sage: F = PolynomialRing(GF(3),'x'); x = F.gen()
sage: S = F.quotient(x^2,'a'); a = S.gen()

I'm trying to find solutions to the following matrix equation:

sage: var('d1,d2,d3')
sage: var('r1,r2,r3')
sage: laplacian = matrix([[d1, -1, 0], [-1, d2, -1], [0, -1, d3]])
sage: rstructure = laplacian.solve_right(vector([0,0,0]))

Now here is the caveat: I want the solutions of rstructure to only be elements of the quotient ring I've defined as S. If you've gotten this far, I'll boil down my main questions below:

(1) What are x=F.gen() and a=S.gen() doing?

(2) I want to see all the possible solutions of r_i, d_i that satisfy the matrix equation with the condition that they are elements of S (the quotient ring that I've defined earlier).

(3) If it is not clear, "rstructure" is an element of (F_3[x]/< x^2 >)^3. What I'm particularly interested in are solutions in (F_p[x])^3. Is computing this even possible for a given prime p?

Thanks for taking the time to read this! Hope to figure this out soon.