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### Relations in a polynomial ring

Suppose SAGE hands me a multivariable polynomial ring in some number of generators. How can I get a list of relations among the generators? In particular, here is the code I'm using:

 rays = [(0,0,1),(1,0,-2),(0,1,-2),(-1,0,-2),(0,-1,-2)] cones = [(1,2,3,4)] Delta = Fan(cones,rays) T = ToricVariety(Delta) R = T.affine_patch(0).coordinate_ring() 

I've tried defining a morphism from a free C-algebra A to my ring which just sends the generators of A to the generators of R, then computing a groebner basis for the ideal of the kernel.

 generators = R.gens() phi = R.hom(generators) 

Sage doesn't seem to let me do this, since kernel isn't implemented for morphisms defined this way. At this point I'm not sure what else to try. I feel like there must be a simpler way to extract relations among generators in a ring, but after spending a couple of days scouring the documentation I can't seem to notice anything relevant. Help is much appreciated!

### Relations in a polynomial ring

Suppose SAGE hands me a multivariable polynomial ring in some number of generators. How can I get a list of relations among the generators? In particular, here is the code I'm using:

 rays = [(0,0,1),(1,0,-2),(0,1,-2),(-1,0,-2),(0,-1,-2)] cones = [(1,2,3,4)] Delta = Fan(cones,rays) T = ToricVariety(Delta) R = T.affine_patch(0).coordinate_ring() 

I've tried defining a morphism from a free C-algebra A R to my ring itself which just sends the generators of A to the generators of R, themselves, then computing a groebner basis for the ideal of the kernel.

 generators = R.gens() phi = R.hom(generators) 

Sage doesn't seem to let me do this, since kernel isn't implemented for morphisms defined this way. At this point I'm not sure what else to try. I feel like there must be a simpler way to extract relations among generators in a ring, but after spending a couple of days scouring the documentation I can't seem to notice anything relevant. Help is much appreciated!

### Relations in a polynomial ring

Suppose SAGE hands me a multivariable polynomial ring in specified by some number of generators. generators, and some relations that I don't know. How can I get a list of relations among the generators? In particular, here is the code I'm using:

 rays = [(0,0,1),(1,0,-2),(0,1,-2),(-1,0,-2),(0,-1,-2)] cones = [(1,2,3,4)] Delta = Fan(cones,rays) T = ToricVariety(Delta) R = T.affine_patch(0).coordinate_ring() 

I've tried defining a morphism from R to itself which just sends the generators to themselves, then computing a groebner basis for the ideal of the kernel.

 generators = R.gens() phi = R.hom(generators) 

Sage doesn't seem to let me do this, since kernel isn't implemented for morphisms defined this way. At this point I'm not sure what else to try. I feel like there must be a simpler way to extract relations among generators in a ring, but after spending a couple of days scouring the documentation I can't seem to notice anything relevant. Help is much appreciated!

 4 No.4 Revision slelievre 14394 ●16 ●135 ●284 http://carva.org/samue...

### Relations in a polynomial ring

Suppose SAGE Sage hands me a multivariable polynomial ring specified by some number of generators, and some relations that I don't know. How can I get a list of relations among the generators? In particular, here is the code I'm using:

 rays = [(0,0,1),(1,0,-2),(0,1,-2),(-1,0,-2),(0,-1,-2)]  [(0,0,1),(1,0,-2),(0,1,-2),(-1,0,-2),(0,-1,-2)]
cones = [(1,2,3,4)]  [(1,2,3,4)]
Delta = Fan(cones,rays)  Fan(cones,rays)
T = ToricVariety(Delta)  ToricVariety(Delta)
R = T.affine_patch(0).coordinate_ring()


I've tried defining a morphism from R to itself which just sends the generators to themselves, then computing a groebner basis for the ideal of the kernel.

 generators = R.gens()  R.gens()
phi = R.hom(generators)


Sage doesn't seem to let me do this, since kernel isn't implemented for morphisms defined this way. At this point I'm not sure what else to try. I feel like there must be a simpler way to extract relations among generators in a ring, but after spending a couple of days scouring the documentation I can't seem to notice anything relevant. Help is much appreciated!