On Thu, Feb 1, 2024 at 5:04 AM Georgi Guninski <ggunin...@gmail.com> wrote: > > On Wed, Jan 31, 2024 at 10:54 PM Dima Pasechnik <dimp...@gmail.com> wrote: > > > > Do you have exactly one (or at most) relation for any pair of variables? > > Can one have i=j ? > > > > Or it's the notation which should be improved? > > > > The number of relations for a pair of variables can be arbitrary.
it's indeed easy to show that a degree-lex Groebner basis will be very small, and only contain terns of degee at most 2. Indeed, S-polynomials you'd get will get its cubic terms reduced. Let us look at the pairs of variables present in your relation, and the connected components of the graph specified by these pairs. For an S-polynomial not to vanish, its leading terms must not be relatively prime, and so at the 1st step you'll compute the S-polynomial of, say, (Ax+B)(Cy+D)=ACxy+ADx+BCy+BD and (Ex+F)(Gz+H) - i.e. have 3 variables involved: x,y,z. (suppose we don't have a relation with yz term) EGz(Ax+B)(Cy+D) - ACy(Ex+F)(Gz+H)= EGADxz + EGBCyz + EGBDz - ACFGyz - ACEHxy - ACFHy = ayz + bx + cy +dz. So we still get just one quadratic term (could be 0), and some more linear terms after reductions. (it's important that for each pair of variables in the linear term there is a relation with a quadratic term involving this pair). It seems that this is basically all that's needed (probably it's something well-known). HTH Dima > i=j appears irrelevant, choose it as you want. > > -- > You received this message because you are subscribed to the Google Groups > "sage-devel" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to sage-devel+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sage-devel/CAGUWgD9AtK43yGeasyiDkyBOvQ_%3DV1Yx930N3%3DPHZ7g723o42g%40mail.gmail.com. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/CAAWYfq15ehGt4j6Rrwg_DbxAQtUFojZ3ZEzOffTKqSp_x53y6Q%40mail.gmail.com.
