Isn't this what you want?
sage: R.<x,y> = QQ[]
sage: phi = R.hom([x,x])
sage: phi
Ring endomorphism of Multivariate Polynomial Ring in x, y over Rational
Field
Defn: x |--> x
y |--> x
sage: phi.kernel()
Ideal (x - y) of Multivariate Polynomial Ring in x, y over Rational Field
On Monday, October 30, 2023 at 6:08:16 PM UTC+9 Dima Pasechnik wrote:
>
>
> On Mon, 30 Oct 2023, 05:57 John H Palmieri, <[email protected]> wrote:
>
>> Does anyone have any tips for how to compute the kernel of a map between
>> polynomial algebras, or for checking whether the map is injective? I have
>> families of such maps involving algebras with many generators. I'm working
>> over GF(2), if that matters. In one example I defined the map phi: R -> S
>> where R has 12 generators, S has 19 generators, and did
>>
>> sage: phi.is_injective()
>>
>> After about 30 hours, Sage quit on me, perhaps running out of memory
>> ("Killed: 9"). An example of the sort of map I'm interested in:
>>
>> sage: phi
>> Ring morphism:
>> From: Multivariate Polynomial Ring in h20, h21, h30, h31, h40, h41, h50
>> over Finite Field of size 2
>> To: Multivariate Polynomial Ring in h20, h21, h30, h31, h40, h41,
>> h50, xi1, xi2, xi3, xi4, xi5 over Finite Field of size 2
>> Defn: h20 |--> h20
>> h21 |--> h21
>> h30 |--> h20*xi1^4 + h21*xi1 + h30
>> h31 |--> h21*xi1^8 + h31
>> h40 |--> h21*xi1^9 + h30*xi1^8 + h20*xi2^4 + h31*xi1
>> h41 |--> h31*xi1^16 + h21*xi2^8
>> h50 |--> h31*xi1^17 + h21*xi1*xi2^8 + h30*xi2^8 + h20*xi3^4
>>
>> Any suggestions?
>>
>
> The standard way to find the kernel of a map
> phi: A->B is to take the
> ring R generated by the gens of A and B and compute the Gröbner basis of
> the ideal I generated by {a-phi(a)|a in gens(A)}, and then
> take the intersection of I with A.
> (for the latter you have to take R with an appropriate order)
>
> The Gröbner basis would be done by Singular.
> Better Gröbner basis routines are available in the msolve spkg.
>
> I'd try using msolve. There are also options such as computing I w.r.t. to
> an "easier" order and then chaniging the order (so-called Gröbner walk),
> they might work better here (it's all more of art than science here)
>
>
>
> HTH
> Dima
>
>
>> --
>> John
>>
>>
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