sage: R.<x,y,z> = PolynomialRing(GF(*7*),*3*)
sage: I = R.ideal(x*y+z+*1*-*1*, x-*1*, y-*5*)
sage: I
Ideal (x*y + z, x - 1, y + 2) of Multivariate Polynomial Ring in x, y, z
over Finite Field of size 7
sage: I.variety()
[{z: 2, y: 5, x: 1}]
On Wednesday, May 11, 2022 at 4:43:23 AM UTC-4 Emmanuel Charpentier wrote:
> Your example has several problems :
>
> 1) You don’t define your polynomial indeterminates ; you should
> Rx.inject_variables().
>
> 2) The syntax you use to substitute values in f is questionable…
>
> 3) f(v) is a polynomial in x0..x9 over GF(7), *not* a symbolic
> expression. Therefore f(v)==1 is *not* a symbolic equation ; it just
> tests if f(v) is equal to 1, which isFalse`. Hence the “error” you get.
>
> 4) solve_modis a function working on *symbolic equations systems*.
> Passing a (list of) polynomial(s) as its first argument will fail. If you
> want to use this function, us it on *symbolic equation systems*.
>
> 5) “Solving” polynomial systems use other methods. Perusing the
> documentation and the source the latter points to is *highly* recommended.
>
> HTH,
>
> Le mardi 10 mai 2022 à 09:40:48 UTC+2, Ha a écrit :
>
>> Hi,
>> I need to solve a system of linear equations [over finite fields] which
>> are obtained from system of polynomials by l substitution of some
>> variables.
>> For example: If we start with F[x1,x2,x3] = x1*x2+x3+1 and let x1 = 1
>> then we get
>> L[x2,x3] = F(1,x2,x3) = x2+x3+1 --> a linear equation.
>> Need to solve L == constant over a finite field Fp.
>>
>> I tried the following method. But no luck.
>>
>> ##--------------------------------------------------------
>> n=10
>> F = GF(7)
>> Rx=PolynomialRing(F,n,'x')
>> X=Rx.gens()
>> f = x2+ x1 * x5 - 1
>> print(f)
>> v=list(X)
>> v[1]=1
>> v[2]=5
>> print(f(v))
>> eqns = [f(v) == 1]
>> sol = solve_mod(eqns, 7)
>> print(sol)
>> ##------------------------------------------------------
>> Getting error:
>>
>> - AttributeError: 'bool' object has no attribute 'lhs'
>>
>>
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