#12802: test containment of ideals in class MPolynomialIdeal
--------------------------------------------------+-------------------------
Reporter: mariah | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-5.1
Component: algebra | Resolution:
Keywords: sd40.5, groebner bases, ideals | Work issues:
Report Upstream: N/A | Reviewers:
Authors: John Perry | Merged in:
Dependencies: | Stopgaps:
--------------------------------------------------+-------------------------
Changes (by {'newvalue': u'John Perry', 'oldvalue': ''}):
* keywords: => sd40.5, groebner bases, ideals
* status: new => needs_review
* author: => John Perry
Old description:
> There seems to be no way to test containment of ideals in the class
> MPolynomialIdeal in {{{sage.rings.polynomial.multi_polynomial_ideal}}}.
> One might expect the comparison operators (e.g. {{{I<J}}} ) to do this,
> but in fact what they do is to compare the Groebner bases as sequences of
> polynomials, which is counterintuitive.
> For example:
>
> {{{
> sage: R.<x,y> = PolynomialRing(QQ)
> sage: I=(x*y)*R; J=(x,y)*R; I<J
> False
> sage: I=(y+1)*R; J=(x,y)*R; I<J
> True
> }}}
>
> This is implemented in the {{{__cmp__}}} method, which is not up to the
> task of doing subset comparison, since {{{__cmp__}}} is only suitable for
> total orderings.
>
> To do it right would seem to require implementing Python's "rich
> comparison" methods, {{{__lt__}}}, {{{__gt__}}}, etc.
>
> For example:
>
> {{{
> from sage.rings.polynomial.multi_polynomial_ideal import MPolynomialIdeal
>
> def IsSubset(I,J):
> for g in I.gens()
> if not g in J: return False
> return True
>
> def IsSuperset(I,J):
> return IsSubset(J,I)
>
> def IsProperSubset(I,J):
> return I!=J and IsSubset(I,J)
>
> def IsProperSuperset(I,J):
> return I!J and IsSuperset(I,J)
>
> setattr(MPolynomialIdeal,'__le__',IsSubset)
> setattr(MPolynomialIdeal,'__lt__',IsProperSubset)
> setattr(MPolynomialIdeal,'__ge__',IsSuperset)
> setattr(MPolynomialIdeal,'__gt__',IsProperSuperset)
> }}}
>
> With these we now get the expected behavior:
>
> {{{
> sage: R.<x,y> = PolynomialRing(QQ)
> sage: I=(x*y)*R; J=(x,y)*R; I<J
> True
> sage: I=(y+1)*R; J=(x,y)*R; I<J
> False
> }}}
New description:
There seems to be no way to test containment of ideals in the class
MPolynomialIdeal in {{{sage.rings.polynomial.multi_polynomial_ideal}}}.
One might expect the comparison operators (e.g. {{{I<J}}} ) to do this,
but in fact what they do is to compare the Groebner bases as sequences of
polynomials, which is counterintuitive.
For example:
{{{
sage: R.<x,y> = PolynomialRing(QQ)
sage: I=(x*y)*R; J=(x,y)*R; I<J
False
sage: I=(y+1)*R; J=(x,y)*R; I<J
True
}}}
This is implemented in the {{{__cmp__}}} method, which is not up to the
task of doing subset comparison, since {{{__cmp__}}} is only suitable for
total orderings.
To do it right would seem to require implementing Python's "rich
comparison" methods, {{{__lt__}}}, {{{__gt__}}}, etc.
For example:
{{{
from sage.rings.polynomial.multi_polynomial_ideal import MPolynomialIdeal
def IsSubset(I,J):
for g in I.gens()
if not g in J: return False
return True
def IsSuperset(I,J):
return IsSubset(J,I)
def IsProperSubset(I,J):
return I!=J and IsSubset(I,J)
def IsProperSuperset(I,J):
return I!J and IsSuperset(I,J)
setattr(MPolynomialIdeal,'__le__',IsSubset)
setattr(MPolynomialIdeal,'__lt__',IsProperSubset)
setattr(MPolynomialIdeal,'__ge__',IsSuperset)
setattr(MPolynomialIdeal,'__gt__',IsProperSuperset)
}}}
With these we now get the expected behavior:
{{{
sage: R.<x,y> = PolynomialRing(QQ)
sage: I=(x*y)*R; J=(x,y)*R; I<J
True
sage: I=(y+1)*R; J=(x,y)*R; I<J
False
}}}
The patch supplied gives a solution via Groebner bases, and also fixes
#12839.
'''Apply''':
* [attachment:trac_12802_and_12839.patch]
--
Comment:
The attached file gives a better check for correctness than even the
proposed solution, using Groebner bases rather than generators alone. The
examples given by mariah produce a correct result on my machine, though
for a doctest I use a slightly different solution which her proposed fix
wouldn't detect.
This patch also resolve #12839, as far as I can tell.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12802#comment:1>
Sage <http://www.sagemath.org>
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