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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!

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!

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!

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()

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