Division of polynomials produces errors when using local orderings

[enriqueartal]

Consider two polynomials f,g in a ring define using a local ordering, e.g.,

R.<x,y>=PolynomialRing(QQ,order='neglex')

If the leading monomial of g is 1 then f/g will produces a wrong result, namely f divided by the independent coefficient of g. Note that in that case g.is_unit() yields True, since for Singular R is not Q[x,y] but the ordering localization (i.e., one can divide by the polynomials whose leading monomial is 1).

PS: It is my first ticket, I apologize for the mistakes.

[SimonKing]

Example:

sage: R.<x,y>=PolynomialRing(QQ,order='neglex')
sage: f = 1+y
sage: g = 1+x
sage: f/g
1 + y
sage: _*g==f
False

[enriqueartal]

As far as I remember, in the definition of the division of two polynomials, it checks first if the denominator is a unit. If it is, it divides each monomial by the indepent coefficient of the polynomial. There may be two possible solutions. The simple one would be to change this behavior for the division. A more permanent one, which was discussed in this ​thread proposes to create a new class of localized rings.

[SimonKing]

Note that this is the code of "_div_" for multivariate polynomial rings:

        cdef poly *p
        cdef MPolynomial_libsingular right = <MPolynomial_libsingular>right_ringelement
        cdef bint is_field = left._parent._base.is_field()
        if p_IsConstant(right._poly, right._parent_ring):
            if is_field:
                singular_polynomial_div_coeff(&p, left._poly, right._poly, right._parent_ring)
                return new_MP(left._parent, p)
            else:
                return left.change_ring(left.base_ring().fraction_field())/right_ringelement
        else:
            return (left._parent).fraction_field()(left,right_ringelement)

So, it is tested with the libsingular function !p_IsConstant whether the divisor is constant, hence, we can divide by the leading coefficient. I bet !p_IsConstant assumes a positive ordering and just tests whether the leading monomial is of degree zero.

[mwest]

I checked the example again and it now works, is this fixed?

sage: R.<x,y>=PolynomialRing(QQ,order='neglex')
sage: f = 1+y
sage: g = 1+x
sage: f/g
(y + 1)/(x + 1)

[tmonteil]

Replying to mwest:

I checked the example again and it now works, is this fixed?

In any case, it deserves a dedicated doctest to prevent future regression.

[mwest]

New commits:

​989c4fc

Added test to verify division works

[lftabera]

This looks good to me. Note that the _div_ has not chaged.

Just one question, would it better if the test appears in src/sage/rings/polynomial/multi_polynomial_element.py? This is where the code in question is executed...

[mwest]

I added the test to the definition of div in src/sage/rings/polynomial/multi_polynomial_element.py are you thinking somewhere else?

[lftabera]

yes, sorry, I was thinking of the method _div_ in src/sage/rings/polynomial/multi_polynomial_libsingular.pyx which is where the actual _div_ is performed for this ring.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b958f36

properly formated ::

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​703ad58

Removed print function from test

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c2c1e9e

removed spacing for assignment within function

[tmonteil]

For reference, there is some documentation about formatting on the developer guide, see ​http://doc.sagemath.org/html/en/developer/coding_basics.html#python-code-style

In particular, you can add spacing between QQ, and order="neglex", and perhaps between the + (though the current code is already readable).

About the ::, just to make things clearer than our previous discussion:

blah

::

it the secure way to write blah followed by a block of code,

blah ::

is equivalent to the previous one.

However,

blah::

is equivalent to

blah:

::

which is therefore equivalent to:

blah: ::

[My personal preference goes to first and third, but it is up to you]

What was wrong in the commit 989c4fc2850c173ea9034f22c7742c614840a770 (comment 10) was

Ensure that :trac:`17638` is fixed.::

wich will print both a dot and a column, but i fear that now you removed both, while it might be good to keep a column.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​8dd8ebe

correcting white spacing

[tmonteil]

It is all good with me, and with the newer location, we can see the test when we do:

sage: R.<x,y> = PolynomialRing(QQ, order='neglex')
sage: f = 1 + y
sage: f._div_??

[vdelecroix]

update milestone 8.3 -> 8.4

[vbraun]

Merge conflict

[slelievre]

Discussed on sage-devel at:

• ​https://groups.google.com/d/topic/sage-devel/nDoUXqpJ6OU/discussion