#19001: Conic morphism creation fails on some base fields
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Reporter: lackermans | Owner:
Type: defect | Status: new
Priority: major | Milestone:
Component: algebraic geometry | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Changes (by mstreng):
* priority: minor => major
Old description:
> The function
> `sage.schemes.plane_conics.con_field.ProjectiveConic_field.hom()` with
> codomain argument `Y` given fails on base fields for which sage doesn't
> know how to simplify fractions of polynomials. Consequently
> `diagonalization()` fails for cases that I need to implement #6881.
>
> My current workaround is to, in case of characteristic 0, convert the
> fraction `q` to a symbolic expression and back. I'm not sure if this is
> mathematically okay, and this still leaves infinite fields of positive
> characteristic.
>
> But is it really necessary for `diagonalization()` to check if the found
> matrix transformation gives the required morphism?
>
> '''Example:'''
>
> {{{
> #!python
> sage: K = FractionField(PolynomialRing(QQ, 't')); (t,) = K.gens(); C =
> Conic(K, [1/2,0, 1, 2, 0, 3])
> sage: C.diagonalization()
> ---------------------------------------------------------------------------
> ValueError Traceback (most recent call
> last)
> <ipython-input-2-c3844e29eb5c> in <module>()
> ----> 1 C.diagonalization()
>
> /home/lennart/sage-build/sage-6.7/local/lib/python2.7/site-
> packages/sage/schemes/plane_conics/con_field.pyc in diagonalization(self,
> names)
> 339 D, T = self.diagonal_matrix()
> 340 con = Conic(D, names = names)
> --> 341 return con, con.hom(T, self), self.hom(T.inverse(), con)
> 342
> 343 def gens(self):
>
> /home/lennart/sage-build/sage-6.7/local/lib/python2.7/site-
> packages/sage/schemes/plane_conics/con_field.pyc in hom(self, x, Y)
> 656 raise ValueError("The matrix x (= %s) does
> not define a " \
> 657 "map from self (= %s) to Y
> (= %s)" % \
> --> 658 (x, self, Y))
> 659 x = Sequence(x*vector(self.ambient_space().gens()))
> 660 return self.Hom(Y)(x, check = False)
>
> ValueError: The matrix x (= [ 1 0 -1]
> [ 0 1 0]
> [ 0 0 1]) does not define a map from self (= Projective Conic Curve
> over Fraction Field of Univariate Polynomial Ring in t over Rational
> Field defined by 1/2*x^2 + 2*y^2 + 5/2*z^2) to Y (= Projective C
> onic Curve over Fraction Field of Univariate Polynomial Ring in t over
> Rational Field defined by 1/2*x^2 + 2*y^2 + x*z + 3*z^2)
> }}}
New description:
The function
`sage.schemes.plane_conics.con_field.ProjectiveConic_field.hom(x, Y)` with
`x` a matrix and codomain argument `Y` given fails on base fields for
which sage does not simplify fractions of multivariate polynomials. It
raises a `ValueError` for perfectly valid input.
{{{
sage: C = Conic(QQ, [1,0,0,1,0,1])
sage: D = Conic(QQ, [1,0,-2,1,0,2])
sage: T = Matrix([[1,0,1],[0,1,0],[0,0,1]])
sage: C.hom(T, D) # works fine over QQ
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 +
z^2
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 -
2*x*z + 2*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + z : y : z)
sage: K = FractionField(PolynomialRing(QQ, 't'))
sage: CK = C.base_extend(K)
sage: DK = D.base_extend(K)
sage: TK = T.base_extend(K)
sage: CK.hom(TK).codomain() == DK # works fine when the codomain is not
specified
True
sage: CK.hom(TK, DK) # fails over K when the codomain is specified
---------------------------------------------------------------------------
ValueError Traceback (most recent call
last)
<ipython-input-65-08845d23f389> in <module>()
----> 1 CK.hom(TK, DK)
/Users/marcostreng/sage_builds/sage/local/lib/python2.7/site-
packages/sage/schemes/plane_conics/con_field.pyc in hom(self, x, Y)
656 raise ValueError("The matrix x (= %s) does not
define a " \
657 "map from self (= %s) to Y
(= %s)" % \
--> 658 (x, self, Y))
659 x = Sequence(x*vector(self.ambient_space().gens()))
660 return self.Hom(Y)(x, check = False)
ValueError: The matrix x (= [1 0 1]
[0 1 0]
[0 0 1]) does not define a map from self (= Projective Conic Curve over
Fraction Field of Univariate Polynomial Ring in t over Rational Field
defined by x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Fraction
Field of Univariate Polynomial Ring in t over Rational Field defined by
x^2 + y^2 + (-2)*x*z + 2*z^2)
}}}
Here is the code that raises the `ValueError`. It should test whether q is
constant, but instead tests whether its numerator and denominator are
constant.
{{{
q = Y.defining_polynomial()/im.defining_polynomial()
if not (q.numerator().is_constant()
and q.denominator().is_constant()):
raise ValueError("The matrix x (= %s) does not define
a " \
"map from self (= %s) to Y (= %s)" %
\
(x, self, Y))
}}}
Here is a less direct example, which shows that consequently
`diagonalization()` fails for such base fields.
'''Example:'''
{{{
#!python
sage: K = FractionField(PolynomialRing(QQ, 't')); (t,) = K.gens(); C =
Conic(K, [1/2,0, 1, 2, 0, 3])
sage: C.diagonalization()
---------------------------------------------------------------------------
ValueError Traceback (most recent call
last)
<ipython-input-2-c3844e29eb5c> in <module>()
----> 1 C.diagonalization()
/home/lennart/sage-build/sage-6.7/local/lib/python2.7/site-
packages/sage/schemes/plane_conics/con_field.pyc in diagonalization(self,
names)
339 D, T = self.diagonal_matrix()
340 con = Conic(D, names = names)
--> 341 return con, con.hom(T, self), self.hom(T.inverse(), con)
342
343 def gens(self):
/home/lennart/sage-build/sage-6.7/local/lib/python2.7/site-
packages/sage/schemes/plane_conics/con_field.pyc in hom(self, x, Y)
656 raise ValueError("The matrix x (= %s) does not
define a " \
657 "map from self (= %s) to Y
(= %s)" % \
--> 658 (x, self, Y))
659 x = Sequence(x*vector(self.ambient_space().gens()))
660 return self.Hom(Y)(x, check = False)
ValueError: The matrix x (= [ 1 0 -1]
[ 0 1 0]
[ 0 0 1]) does not define a map from self (= Projective Conic Curve over
Fraction Field of Univariate Polynomial Ring in t over Rational Field
defined by 1/2*x^2 + 2*y^2 + 5/2*z^2) to Y (= Projective C
onic Curve over Fraction Field of Univariate Polynomial Ring in t over
Rational Field defined by 1/2*x^2 + 2*y^2 + x*z + 3*z^2)
}}}
--
Comment:
This is a serious bug in `hom`, giving incorrect information to the user.
That needs to be fixed, independently of what it means for the
diagonalization method, and for all fields once and for all. I removed the
workaround (which only works for certain fields) from the ticket
description.
As a temporary workaround: avoid giving `Y` to `hom` in conics, and use
`diagonal_matrix()` directly instead of `diagonalization()`.
--
Ticket URL: <http://trac.sagemath.org/ticket/19001#comment:3>
Sage <http://www.sagemath.org>
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