Finding all admissible ideals with SAGE

asked 7 years ago

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updated 7 years ago

Let A=k[x,y,z] for a finite field k and let J=(x,y,z) be the ideal generated by x,y and z. Call an Ideal I of A r-admissible in case Jr is contained in I and I is contained in J2. Can SAGE find all 4-admissible ideals for a given finite field (lets say with 2,3 or 5 elements for a start)? This is one of the easiest special cases of a more general problem, which is probably too hard and too slow for todays computer. But maybe SAGE can do it in principle? Note that this is a finite problem as it is equivalent to finding all ideals in A/J4 that are contained in J2/J4, which are all finite rings. (More general it would be more interesting to do the same with A replaced by the non-commutative polynomial ring in x,y,z, which is the quiver algebra with 1 point and 3 loops)

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