multivariate polynomials over residue fields of number fields are broken

[e13df781-8644-42aa-9d66-1e8d332e25bb]

This example from Genya Zaytman:

sage: F1.<u> = NumberField(x^6 + 6*x^5 + 124*x^4 + 452*x^3 + 4336*x^2 + 8200*x + 42316)
sage: reduct_id = F1.factor_integer(47)[0][0]
sage: Rf = F1.residue_field(reduct_id)   # = GF(47^3)
sage: R1.<X,Y> = PolynomialRing(Rf)
sage: ubar = Rf(u)
sage: I = ideal([ubar*X+Y])
sage: I.groebner_basis()
[boom]

Basically all we're doing is working with polynomials over a finite field. Perhaps the singular interface can't handle the way the field is presented, or something like that.

Component: commutative algebra

Keywords: residue field multivariate prime groebner basis

Issue created by migration from https://trac.sagemath.org/ticket/2789

[93749b3a-c0a4-47a7-b178-004ba08b0b97]

comment:2

This now works without major changes... but I found a show stopper bug. It had to do with hashing "large" residue fields, meaning large characteristic residue fields. The hash method tried to hash an ideal of the residue field itself, which tried to hash its parent, leading to an infinite loop.

This means that at no time has a residue field with cardinality a very large prime been created in Sage!

I've added the obvious .ideal() method, tested the hashing and construction for larger examples, and added doctests showing Groebner basis computations.

[93749b3a-c0a4-47a7-b178-004ba08b0b97]

Changed keywords from none to residue field multivariate prime groebner basis

[malb]

Attachment: trac_2789.patch.gz

Attachment: trac_2789.2.patch.gz

replaces some line numbers in verbose output by ...

[85eec1a4-3d04-4b4d-b711-d4db03337c41]

comment:4

Merged in Sage 3.3.alpha1