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All the solutions of a polynomial system over a finite ring

asked 2022-06-11 12:47:46 +0200

updated 2022-10-15 13:29:01 +0200

FrédéricC gravatar image

Let $E$ be a finite list of polynomial equations with variables $t_0, \dots, t_r$ over the finite ring of integers modulo $n$ (for some $n$). We can compute the Groebner basis as follows:


Usually, over a field and if Id.dimension()=0 then we can get all the solutions by Id.variety(). But here the dimension is not necessarily $0$ and moreover it is not over a field but a ring, the finite ring of integers modulo $n$. By finiteness of the ring, there still have finitely many solutions. How to get them all?

Example: $n=248832$, $r=10$ and:

sage: G
[t4*t7, t7*t8, t7*t9, t0, t1, t2, t3 + 248831*t4, 2*t4, t5 + 248783*t7, t6 + 11*t7, 9*t7, 4*t8 + 248828*t9, 16*t9]
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In your example, all non-linear polynomials (like t4*t7) are reducible and furthermore are products of linear factors. So, it's enough to iterate over their factors, making them equal to divisors of zero, to obtain a number of systems of linear equations, which can be solved via integer lattices like explained in

Max Alekseyev gravatar imageMax Alekseyev ( 2022-06-11 14:14:14 +0200 )edit

@Max Alekseyev Is it working as well for the finite ring of integers modulo $n$?

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2022-06-11 15:53:24 +0200 )edit

The standard approach to modeling "modulo $n$" is by adding a term $n\cdot z_i$ to the $i$-th equation (where $z_i$ are new integer variables) and considering the resulting system over the integers.

Max Alekseyev gravatar imageMax Alekseyev ( 2022-06-11 17:10:07 +0200 )edit

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answered 2022-06-11 19:10:32 +0200

Max Alekseyev gravatar image

updated 2022-07-01 18:57:22 +0200

In fact, my custom implementation of variety() method from other answer works fine here, provided two things:

  • a custom implementation of roots() method that supports polynomials with the content non-coprime to the ring modulus;
  • a correct implementation of interreduced_basis()method avoiding the bug

It finds all 1152 solutions within a few seconds.

import itertools
from sage.rings.finite_rings.integer_mod_ring import crt as mycrt

def myinterreduced_basis(J):
    B = list(J.gens())
    R = J.ring()
    while True:
        updated = False
        for i in range(len(B)):
            r = R.ideal(list(set(B)-{B[i]})).reduce(B[i])
            if r!=B[i]:
                B[i] = r
                updated = True
        if not updated:
            return B
        B = [b for b in B if b!=0]

def myroots(p):
    R = p.base_ring()
    if not isinstance(R,
        yield from p.roots(multiplicities=False)

    m =
    pz = p.change_ring(ZZ)
    g = gcd(gcd(pz.coefficients()),m)
    mg = m//g

    roots = (pz/g).roots(ring=Integers(mg),multiplicities=False)  # roots modulo mg = m/g

    # embedding roots modulo mg to modulo m=mg*g
    for rr in itertools.product( *( [Mod(r*q^valuation(mg,q),q^k) for r in range(q^valuation(g,q))] for q,k in factor(m) ) ):
        r2 = mycrt(rr)
        yield from (r2 + ZZ(r) for r in roots)

In fact, computing the interreduced basis of G immediately eliminates quadratic terms, and so the problem can be reduced solving linear equations right away as explained in the comments. This way it takes just a fraction of a second to get all the solutions.

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I solved the example by hand, and I found $1152$ solution (whereas you found $144$ ones). The problem may come from the quadratic equations: for all $1152$ solutions, t4*t7, t7*t8 and t7*t9 are always equal to $0$ modulo $n$ (unless I am mistaken).

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2022-06-12 06:05:01 +0200 )edit

Here is the reason for missing solutions:

Max Alekseyev gravatar imageMax Alekseyev ( 2022-06-12 20:12:49 +0200 )edit

But you did not use interreduced_basis. I hope that the use of groebner_basis on non-field (say Integers(n)) does not produce wrong results.

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2022-06-13 04:46:36 +0200 )edit

It is used in myvariety function. Replacing it with groebner_basis may help, but will be slower.

Max Alekseyev gravatar imageMax Alekseyev ( 2022-06-13 05:13:20 +0200 )edit

I've updated my answer accordingly and all 1152 solutions are in place. I also confirm that all t4*t7, t7*t8 and t7*t9 are implied to be zero by the other terms as they are eliminated in the interreduced basis. Their presence in the Groebner basis seems to indicate that there is also a bug in .groebner_basis() method.

Max Alekseyev gravatar imageMax Alekseyev ( 2022-06-13 17:08:23 +0200 )edit

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Asked: 2022-06-11 12:47:46 +0200

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Last updated: Jul 01 '22