Solutions to Equations over different fields

Hello, suppose I want to solve a system of equations

w,x,y,z = SR.var('w, x, y, z')
eqn1=2*w*y^2 + y^2*z + z^3==0
eqn2=4*x^3 - 2*x*y^2 - y^3==0
eqn3=2*w^2*y - 2*x^2*y - 3*x*y^2 + 2*w*y*z==0
eqn4=w*y^2 + 3*w*z^2==0
sol = solve([eqn1, eqn2, eqn3, eqn4], w,x,y,z, solution_dict=True)
sol


This gives me solutions over complex numbers I believe? How can I ask for solutions over say a finite field. Let's make it more simple and say the field is a prime and not a prime power. Hence, just reducing modulo p. As you all know, it is possible more solutions arise when moving to finite fields.

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Do not use the symbolic ring SR, but a polynomial ring over a finite field.

( 2021-07-29 09:12:13 +0100 )edit

def buildit(p):
if not p.is_prime():
raise DomainError("p must be prime !")
R1=PolynomialRing(Zmod(p),"w, x, y, z")
return [R1(u.lhs()) for u in (eqn1, eqn2, eqn3, eqn4)]

def dimres(p):
S1=buildit(p)
J1=S1[0].parent().ideal(*S1)
return J1.dimension()

sage: dimres(2)
2

sage: %time all([v==1 for v in [dimres(u) for u in prime_range(3,100000)]])
CPU times: user 2min 20s, sys: 841 ms, total: 2min 21s
Wall time: 2min 22s
True


BTW :

sage: sol
[{w: r1, x: 0, y: 0, z: 0}]


HTH,

( 2021-07-29 12:43:18 +0100 )edit

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Here is an example for the finite field GF(3)

Let us first define the field:

sage: p = 3
sage: K = GF(p)
sage: K
Finite Field of size 3


Then, let us define the polynomial ring:

sage: R.<w,x,y,z> = K[]
sage: R
Multivariate Polynomial Ring in w, x, y, z over Finite Field of size 3


Then, let us define the 4 polynomials which have to vanish:

sage: eqn1 = 2*w*y^2 + y^2*z + z^3
sage: eqn2 = 4*x^3 - 2*x*y^2 - y^3
sage: eqn3 = 2*w^2*y - 2*x^2*y - 3*x*y^2 + 2*w*y*z
sage: eqn4 = w*y^2 + 3*w*z^2


Now, let us define the ideal generated by those polynomials:

sage: I = R.ideal([eqn1, eqn2, eqn3, eqn4])
sage: I
Ideal (-w*y^2 + y^2*z + z^3, x^3 + x*y^2 - y^3, -w^2*y + x^2*y - w*y*z, w*y^2) of Multivariate Polynomial Ring in w, x, y, z over Finite Field of size 3


Unfortunately, this ideal has dimension 1:

sage: I.dimension()
1


Hence, Sage is not able to find its variety (which corresponds to the set of common zeroes of the 4 polynomials above):

sage: I.variety()
ValueError: The dimension of the ideal is 1, but it should be 0


Hovewer there is a trick : on GF(p) there fro the Frobenius, we know that the polynomial x^p-x vanishes, hence, since we are looking to the solutions within GF(p)^4, we can add those polynomials to the generators of the ideal:

sage: fw = w^p - w
sage: fx = x^p - x
sage: fy = y^p - y
sage: fz = z^p - z

sage: J = R.ideal([eqn1, eqn2, eqn3, eqn4, fw, fx, fx, fz])
sage: J.dimension()
0
sage: J.variety()
[{z: 0, y: 0, x: 0, w: 0}, {z: 0, y: 0, x: 0, w: 2}, {z: 0, y: 0, x: 0, w: 1}]

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Hello take p=5, what would it mean if the dimension of variety is still one after adding in the Frobenius equations? For example, take eqn1= x^2y + xy^2 + 2wz^2, eqn2= 2wxy + wy^2 + 2xy^2 + z^3, eqn 3=wx^2 + 2wxy + 2x^2y, and eqn4= 2w^2z + 3xz^2 + 4*z^3? If I run the thing above, and hit J.variety(), it is still dimension 1. Does this mean there can possibly be solutions but Sage cannot compute it?

( 2021-07-30 17:48:28 +0100 )edit

Indeed, Sage is not able to solve this system with the variety method, however, it should be possible do brute force over GF(5)^4.

( 2021-07-30 22:41:32 +0100 )edit

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Last updated: Jul 30 '21