So Sage doesn't already use Gröbner bases when computing kernels of such maps? Okay, I'll try that.
Now that I've looked at the code a little bit, I see that `phi.is_injective()` just calls `phi.kernel()` and checks whether it's zero. I was hoping that there was something more clever: if I want to know whether something is injective, I only care whether the kernel is zero, not precisely what the kernel is. By the way, this struck me as odd: sage: phi Ring morphism: From: Multivariate Polynomial Ring in h20, h21, h30, h31, h40, h41 over Finite Field of size 2 To: Multivariate Polynomial Ring in h20, h21, h30, h31, xi1, xi2, xi3, xi4 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 sage: %time phi.is_injective() CPU times: user 7.32 s, sys: 58.7 ms, total: 7.38 s Wall time: 7.44 s True Note that phi doesn't do anything very interesting with h20 and h21: it sends each of those to itself. If I remove them from the domain (I need to keep them in the codomain because they're involved in other terms), the computation is slower: sage: phi Ring morphism: From: Multivariate Polynomial Ring in h30, h31, h40, h41 over Finite Field of size 2 To: Multivariate Polynomial Ring in h20, h21, h30, h31, xi1, xi2, xi3, xi4 over Finite Field of size 2 Defn: 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 sage: %time phi.is_injective() CPU times: user 15.7 s, sys: 101 ms, total: 15.8 s Wall time: 15.9 s True I've seen this on two different machines: roughly double the time to do the second computation. 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 > >>> > >>> > >>> -- > >>> You received this message because you are subscribed to the Google > Groups "sage-support" group. > >>> To unsubscribe from this group and stop receiving emails from it, send > an email to sage-support...@googlegroups.com. > >>> To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-support/97318b8e-f4c9-4af3-a8ff-b901a4f2c971n%40googlegroups.com > . > > > > -- > > You received this message because you are subscribed to the Google > Groups "sage-support" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to sage-support...@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-support/487bf189-fce6-4b6b-9752-178602ff9808n%40googlegroups.com > . > -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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