Polynomial division without remainder

[martin....@gmx.net]

Working in a stack of multivariate polynomial rings, how can I compute the quotient of two polynomials in those cases where I know the remainder to be zero?

Reading the docs I found two likely approaches, but neither seems to work as I'd have hoped. See below for error messages.

Example:

sage: PR1.<a,b>=QQ[]
sage: PR2.<x,y>=PR1[]
sage: n=(x-y)*(x+3*y)
sage: d=(x-y)
sage: n/d
(x^2 + 2*x*y + (-3)*y^2)/(x - y)
sage: PR2(n/d)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-6-cc8d100cc210> in <module>()
----> 1 PR2(n/d)

…/sage/rings/polynomial/multi_polynomial_ring.pyc in __call__(self, x, check)
    449                 return x.numerator()
    450             else:
--> 451                 raise TypeError, "unable to coerce since the denominator is not 1"
    452
    453         elif is_SingularElement(x) and self._has_singular:

TypeError: unable to coerce since the denominator is not 1
sage: n.quo_rem(d)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-7-d17781c77d90> in <module>()
----> 1 n.quo_rem(d)

…/sage/structure/element.so in sage.structure.element.NamedBinopMethod.__call__ (sage/structure/element.c:24924)()

…/sage/rings/polynomial/multi_polynomial_element.pyc in quo_rem(self, right)
   1730             return self.quo_rem(right)  # this looks like recursion, but, in fact, it may be that self, right are a totally new composite type
   1731         R = self.parent()
-> 1732         R._singular_().set_ring()
   1733         X = self._singular_().division(right._singular_())
   1734         return R(X[1][1,1]), R(X[2][1])

…/sage/rings/polynomial/polynomial_singular_interface.pyc in _singular_(self, singular)
    213             return R
    214         except (AttributeError, ValueError):
--> 215             return self._singular_init_(singular)
    216
    217     def _singular_init_(self, singular=singular_default):

…/sage/rings/polynomial/polynomial_singular_interface.pyc in _singular_init_(self, singular)
    229         """
    230         if not can_convert_to_singular(self):
--> 231             raise TypeError, "no conversion of this ring to a Singular ring defined"
    232
    233         if self.ngens()==1:

TypeError: no conversion of this ring to a Singular ring defined

[Nils Bruin]

On Monday, March 24, 2014 7:56:33 AM UTC-7, martin....@gmx.net wrote:

Working in a stack of multivariate polynomial rings, how can I compute the quotient of two polynomials in those cases where I know the remainder to be zero?

Reading the docs I found two likely approaches, but neither seems to work as I'd have hoped. See below for error messages.

Example:

sage: PR1.<a,b>=QQ[]
sage: PR2.<x,y>=PR1[]
sage: n=(x-y)*(x+3*y)
sage: d=(x-y)

 

You  can do this by manually converting into a singular-compatible ring and back:

sage: S=QQ['a,b,x,y']
sage: PR2(S(n)//S(d))
x + 3*y

This is one place where the fact that sage considers variable names as significant, comes in  handy: it can figure out how to map PR2 to S and back.