On Mon, Oct 30, 2023 at 12:54 PM Kwankyu <[email protected]> wrote:
>
> 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
that's the kernel of the endomorphism phi of R.
John's question is a bit different, and it will require
finding the intersection of such an ideal with the domain of his map.
His R=F_2[h20,...,h50,xi1,...,xi5] and phi induces an endomorphism of
R with the kernel <h_ij-phi(h_ij) I i,j in [(2,0),..,(5,0)]>.
Then phi is injective iff the intersection of this ideal with
F_2[h20,...,h50]={0}.
And this needs a Grobner basis computation.
By the way, using
h30 |--> h20*xi1^4 + h21*xi1 + h30
h31 |--> h21*xi1^8 + h31
one can split the problem into cases
1) xi1=0
2) h21=h20=0
(but perhaps it's only specific to this particular example)
>
> 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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