On Mon, 30 Oct 2023, 20:50 Dima Pasechnik, <[email protected]> wrote:
> > > On Mon, 30 Oct 2023, 20:25 John H Palmieri, <[email protected]> > 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 <[email protected]> >> 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 <[email protected]> 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, <[email protected]> >> 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 [email protected]. >> >> >>> 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 [email protected]. >> >> > 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 [email protected]. >> > 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. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/sage-support/e301dc26-c7b5-4f82-a74a-57eaf0769d0dn%40googlegroups.com >> <https://groups.google.com/d/msgid/sage-support/e301dc26-c7b5-4f82-a74a-57eaf0769d0dn%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> > -- You received this message because you are subscribed to the Google Groups "sage-support" group. 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