Hi John, I tried running msolve on your input (more precisely, converting it into the problem of finding the Grobner basis w.r.t. to the elimination order, as I explained), and I see that it's an injective map.
Computation takes about 3 minutes on an old laptop.
Specifically, I merely run msolve at the command line, on the attached
input file x2,
and output file y2
as follows. Here -v 2 is verbosity level, -t 4 means 4 threads, -e 10
means eliminate
the 1st 10 variables.
(I renamed the variables in (almost) alphabetical order, as I thought
msolve cannot
handle variable names like you have, but, no it wasn't actually necessary)
As far as I can see, the outputted basis does not have elements which only
have the varibles in the domain of your map (in x2 they are p, q, r,
s, t, u, v),
meaning that the map has no nontrivial kernel.
$ msolve -f x2 -o y2 -v 2 -g 2 -t 4 -e 10
On Mon, Oct 30, 2023 at 9:02 PM Dima Pasechnik <dimp...@gmail.com> wrote:
>
>
>
> On Mon, 30 Oct 2023, 20:50 Dima Pasechnik, <dimp...@gmail.com> wrote:
>>
>>
>>
>> On Mon, 30 Oct 2023, 20:25 John H Palmieri, <jhpalmier...@gmail.com> wrote:
>>>
>>>
>>>
>>> On Monday, October 30, 2023 at 12:28:18 PM UTC-7 Dima Pasechnik wrote:
>>>
>>> On Mon, Oct 30, 2023 at 5:04 PM John H Palmieri <jhpalm...@gmail.com> wrote:
>>> >
>>> > Are endomorphisms better to work with? I might be able to extend my map
>>> > to an endomorphism of the larger ring, if that would make the computation
>>> > easier. Probably just send xi1 -> xi1, xi2 -> xi2, etc.
>>>
>>> these are "already there", as if phi is an endomorphism then ker(phi)
>>> is generated by a-phi(a) - so
>>> whenever phi(a)=a this reduces to 0.
>>>
>>>
>>>
>>> I don't understand this. Define phi: k[x,y,z] -> k[x,y,z] by
>>>
>>> x -> y
>>> y -> z
>>> z -> 0
>>>
>>> Then x - phi(x) = x - y is not in the kernel. What do you mean by
>>> "ker(phi) is generated by a-phi(a)"?
>>
>>
>> OK, sorry, we're getting confused.
>> You are interested in checking whether phi, like this:
>> phi: k[x,y,z] -> k[x',y',z']
>> x->y', y->z', z->0
>>
>> is an epimorphism. This is the same as saying that the kernel of phi, which
>> is the intersection of the ideal
>> (x-y', y-z', z) in k[x,y,z,x',y',z'] with k[x,y,z], is trivial, i.e., zero.
>> (in this case it's not trivial, it contains z).
>>
>> So yes, one can think of phi as inducing an endomorphism of
>> k[x,y,z,x',y',z'], of a special kind.
>> How this relates to endomorphisms of k[x,y,z], I don't know.
>
>
> if phi had a fixed point, like x->cx with c in k^*,
> then one could be a bit more economic, and do not introduce x' (and the
> corresponding ideal generator x-cx'), but immediately substitute it with x/c.
>
>
>>
>>
>>>
>>>
>>>
>>> >
>>> > On Monday, October 30, 2023 at 7:14:16 AM UTC-7 Dima Pasechnik wrote:
>>> >>
>>> >> On Mon, Oct 30, 2023 at 12:54 PM Kwankyu <ekwa...@gmail.com> 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, <jhpalm...@gmail.com>
>>> >> >> 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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x2
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