Opened 5 years ago
Last modified 2 weeks ago
#23621 new defect
Fix quotients of univariate polynomial rings over ZZ
Reported by: | Maarten Derickx | Owned by: | |
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Priority: | critical | Milestone: | sage-9.8 |
Component: | commutative algebra | Keywords: | ideal |
Cc: | Samuel Lelièvre | Merged in: | |
Authors: | Reviewers: | ||
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
The quotient of ZZ[x]
by the ideal (x, 2)
works fine using a multivariate polynomial ring:
sage: R.<x> = PolynomialRing(ZZ, 1) sage: I = R.ideal([x, 2]) sage: I Ideal (x, 2) of Multivariate Polynomial Ring in x over Integer Ring sage: S = R.quo(I) sage: [[S(a) == S(b) for b in (0, 2, x)] for a in (0, 2, x)] [[True, True, True], [True, True, True], [True, True, True]]
but it fails using a univariate polynomial ring, returning mathematically wrong answers:
sage: R.<x> = ZZ[] sage: I = R.ideal([x, 2]) sage: I Ideal (x, 2) of Univariate Polynomial Ring in x over Integer Ring sage: S = R.quo(I) sage: sage: [[S(a) == S(b) for b in (0, 2, x)] for a in (0, 2, x)] [[True, False, False], [False, True, False], [False, False, True]]
Expected:
[[True, True, True], [True, True, True], [True, True, True]]
Change History (12)
comment:1 Changed 5 years ago by
comment:2 Changed 5 years ago by
To solve this, I think one needs to implement a new class for ideals in ZZ['x']
and set _ideal_class_
appropriately on R
. Of course, one can argue that the default behavior of the reduce
method on a generic ideal should be to raise an error rather than just return the input unchanged.
comment:3 Changed 5 years ago by
Yeah I totally agree that it should raise an error, because this implementation does not satisfy the assumption on reduce in other parts of the code. For example this is an excerpt from sage/rings/quotient_ring.py
.
The only requirement is that the two-sided ideal `I` provides a ``reduce`` method so that ``I.reduce(x)`` is the normal form of an element `x` with respect to `I` (i.e., we have ``I.reduce(x) == I.reduce(y)`` if `x-y \in I`, and ``x - I.reduce(x) in I``). H
And I think that this is a logic requirement to put on the reduce method.
comment:4 Changed 5 years ago by
Ok there are quite a few doctest failures. If I just make it raise an error. Ironically the first failure is
sage: sage: MS = MatrixSpace(GF(5),2,2) ....: sage: I = MS*[MS.0*MS.1,MS.2+MS.3]*MS ....: sage: Q = MS.quo(I) ....: sage: Q.0*Q.1 # indirect doctest ....: --------------------------------------------------------------------------- NotImplementedError Traceback (most recent call last) ... NotImplementedError: reduce not implemented for Twosided Ideal ( [0 1] [0 0], [0 0] [1 1] ) of Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 5
which was added to test that #11068 is fixed, the ticket where the above text about "The only requirement is that the two-sided ideal I..." comes from.
comment:5 Changed 5 years ago by
The second failure points at #13999 of which this ticket basically is a dupe.
comment:6 Changed 5 years ago by
All failures will probably be fixed if these three tests pass
sage: MS = MatrixSpace(GF(5),2,2) sage: I = MS*[MS.0*MS.1,MS.2+MS.3]*MS sage: Q = MS.quo(I) sage: Q.0*Q.1 # indirect doctest [0 1] [0 0]
sage: R.<x> = PolynomialRing(ZZ) sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2]) sage: S = R.quotient_ring(I); sage: TestSuite(S).run(skip=['_test_nonzero_equal', '_test_elements', '_test_zero'])
sage: S = SteenrodAlgebra(2) sage: I = S*[S.0+S.1]*S sage: Q = S.quo(I) sage: Q.0 Sq(1)
I consider all three of them bugs, so this strengthens my believe that it is better to raise a NotImplementedError?.
comment:7 Changed 5 years ago by
I think that all the matrix space examples will not give any interesting doctest, since matrix rings over fields are simple and hence there are no two sided ideals. Although this means that the reduce function is very easy to implement! I don't know enough about Steenrod algebra's in order to create a meaningful reduce method.
comment:8 Changed 10 months ago by
Description: | modified (diff) |
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comment:9 Changed 10 months ago by
Cc: | Samuel Lelièvre added |
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Description: | modified (diff) |
Keywords: | ideal added |
Milestone: | sage-8.1 → sage-9.5 |
Summary: | Quotients of univariate polynomial rings over ZZ return mathematical incorrect answers → Fix quotients of univariate polynomial rings over ZZ |
comment:10 Changed 9 months ago by
Milestone: | sage-9.5 → sage-9.6 |
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comment:11 Changed 5 months ago by
Milestone: | sage-9.6 → sage-9.7 |
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comment:12 Changed 2 weeks ago by
Milestone: | sage-9.7 → sage-9.8 |
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The problem is that
I
is just a generic ideal and doesn't implement areduce
method.