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.
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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