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.
>
>
> >
> > 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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