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)"? > > 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. > 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/0ddf54b3-1778-4fca-932c-bb5521963db2n%40googlegroups.com. -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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