Thanks, Dima. That works for me, too, and it's much faster than Sage was.
Now I'm trying some bigger examples...
On Monday, October 30, 2023 at 3:56:00 PM UTC-7 Dima Pasechnik wrote:
> 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 <dim...@gmail.com> wrote:
> >
> >
> >
> > On Mon, 30 Oct 2023, 20:50 Dima Pasechnik, <dim...@gmail.com> wrote:
> >>
> >>
> >>
> >> On Mon, 30 Oct 2023, 20:25 John H Palmieri, <jhpalm...@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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