![]() | 1 | initial version |
We define the ring and the ideal:
sage: R.<x1,x2,x3> = GF(2)[]
sage: R
Multivariate Polynomial Ring in x1, x2, x3 over Finite Field of size 2
sage: I = R.ideal(x1*x2*x3-1, x2-x1, x1-1)
We compute the groebner basis of the ideal and we realize that these are exactly the right hand side of the three given equations:
sage: I.groebner_basis()
[x1 + 1, x2 + 1, x3 + 1]
sage: [x1-1, x2-1, x3-1]
[x1 + 1, x2 + 1, x3 + 1]
Thus, finding a solution to the 3 equations is equivalent to finding a solution to the following:
x1w2+x2w3∈I,x2w1+x3w3∈I,x1w1+x3w2∈I
which holds by definition of ideal for all elements w1,w2,w3 in the ideal I. Now, this does not include all solutions, for example:
sage: x1*x2 + x2*x3 in I
True
sage: x2 in I
False
sage: x3 in I
False
I stop here. Maybe you can continue finding all solutions using the groebner basis of the ideal computed above.
![]() | 2 | No.2 Revision |
We define the ring and the ideal:
sage: R.<x1,x2,x3> = GF(2)[]
sage: R
Multivariate Polynomial Ring in x1, x2, x3 over Finite Field of size 2
sage: I = R.ideal(x1*x2*x3-1, x2-x1, x1-1)
We compute the groebner basis of the ideal and we realize that these are exactly the right hand side of the three given equations:
sage: I.groebner_basis()
[x1 + 1, x2 + 1, x3 + 1]
sage: [x1-1, x2-1, x3-1]
[x1 + 1, x2 + 1, x3 + 1]
Thus, finding a solution to the 3 equations is equivalent to finding a solution to the following:
x1w2+x2w3∈I,x2w1+x3w3∈I,x1w1+x3w2∈I
which holds by definition of ideal for all elements w1,w2,w3 in the ideal I. I which solves the question as it was requested that the wi are in the ideal. Now, notice that this does not include all solutions, for example:
sage: x1*x2 + x2*x3 w2 = x2; w3 = x3
sage: w2 in I, w3 in I
(False, False)
sage: x1*w2 + x2*w3 in I
True
sage: x2 in I
False
sage: x3 in I
False
I stop here. Maybe you can continue finding all solutions using the groebner basis of the ideal computed above.