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Groebner bases are somehow overkill: they are used for polynomial systems (involving equations like $ax^3y^4+by^5z+cxz+d=0$).

Here your system is linear, so linear algebra (matrix computations) is definitely the appropriate tool. It is much faster, can handle larger system, etc.

If you want to work modulo some prime p, you should define your matrices over the finite field GF(p):

sage: matrix(GF(5), [[1, 2, 4, 2], [2, 4, 4, 0], [0, 4, 4, 1], [2, 0, 0, 2]])


If you really want to use ideals, you can easily define your polynomials over GF(p), here is an example:

sage: R.<x,y,z> = PolynomialRing(GF(5)) ; R
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
sage: P = 2*x+3*y+5*z+2
sage: Q = 2*x+y+5*z+2
sage: S = x+y+z
sage: I = R.ideal([P,Q,S]) ; I
Ideal (2*x - 2*y + 2, 2*x + y + 2, x + y + z) of Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
sage: I.variety()
[{y: 0, z: 1, x: 4}]


Note that when the variety is not zero dimensional, you might encounter issues.

sage: I = R.ideal([P,Q])
sage: I.variety()
ValueError: The dimension of the ideal is 1, but it should be 0


In linear words, this case corresponds to to having a non-trivial kernel, a case that is very well handled with matrices, see the following list of examples:

• https://ask.sagemath.org/question/37042/how-to-find-solution-to-the-following-matrix/
• https://ask.sagemath.org/question/36948/how-to-simplify-solve-result-r_i-variables/
• https://ask.sagemath.org/question/35227/basic-question-about-annihilator/
• https://ask.sagemath.org/question/31009/linear-equations-with-infinite-solutions/
• https://ask.sagemath.org/question/31163/enumerate-all-solutions-to-linear-system-over-finite-field/
• https://ask.sagemath.org/question/32145/solving-system-of-linear-equations-over-gf2/
• https://ask.sagemath.org/question/33915/linear-algebra-in-finite-fields-goppa-codes/