On Mon, Oct 30, 2023 at 12:54 PM Kwankyu <ekwan...@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+unsubscr...@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+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/CAAWYfq3h5d_swFG%2B83EtuDdA15KFuXrekmzeUX4WFU3-H8jfAA%40mail.gmail.com.