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.

CC: @simon-king-jena @sagetrac-jakobkroeker @slel

Component: commutative algebra

Keywords: Singular, days94

Author: Mckenzie West

Branch/Commit: 8dd8ebe

Reviewer: Luis Felipe Tabera Alonso, Thierry Monteil

Issue created by migration from https://trac.sagemath.org/ticket/17638

[simon-king-jena]

comment:2

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]

comment:3

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.

[simon-king-jena]

comment:4

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.

[sagetrac-jakobkroeker]

Changed keywords from none to Singular

[mckenziewest]

comment:7

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)

[sagetrac-tmonteil]

comment:8

Replying to @mckenziewest:

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

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

[mckenziewest]

Branch: u/mwest/division_of_polynomials_produces_errors_when_using_local_orderings

[mckenziewest]

Changed keywords from Singular to Singular, days94

[mckenziewest]

Commit: 989c4fc