#6496: functions numerator() and denominator() for multivariate polynomials
-------------------------+--------------------------------------------------
Reporter: mvngu | Owner: tbd
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.1.1
Component: algebra | Keywords: numerator, denominator, multivariate
polynomials
Reviewer: | Author: lftabera
Merged: |
-------------------------+--------------------------------------------------
Changes (by newvalueoldvalue):
* author: => lftabera
Comment:
I sent the original patch. I have realized that SAGE does neither
implement numerator and denominator for univariate polynomials in general.
I have open a ticket with patches in #6541 so maybe is better to wait
until that patch is accepted to review this one.
I have written a new patch trying to be compatible to the univariate case.
a new function denominator in MPolynomial:
1.- First it tries to compute the lcm of the denominators of the
coefficients. Returns this computation as result.
2.- If it cannot compute the previous denominator, returns a generic
denominator. self.parent().one_element()
a new function numerator in MPolynomial:
returns self * self.denominator()
In the univariate case there is a special case, the numerator of a
polynomial with rational coefficients QQ['x'] in a polynomial with integer
coefficients ZZ['x'] So I have added another patch to
MPolynomial_libsingular.
3.- If the base ring of the polynomial is the field of rational numbers,
the numerator is coerced to the ring of multivariate polynomials over the
integers, with the same variables and same ordering as self.parent()
{{{
diff -r ca1f31d6f6bf sage/rings/polynomial/multi_polynomial.pyx
--- a/sage/rings/polynomial/multi_polynomial.pyx Thu Jul 09
15:14:36 2009 -0700
+++ b/sage/rings/polynomial/multi_polynomial.pyx Wed Jul 15
20:51:20 2009 -0700
@@ -980,8 +980,136 @@
from sage.structure.factorization import Factorization
return Factorization(v, unit)
-
-
+ def denominator(self):
+ """
+ Return a denominator of self.
+
+ First, the lcm of the denominators of the entries of self
+ is computed and returned. If this computation fails, the
+ unit of the parent of self is returned.
+
+ Note that some subclases may implement its own denominator
+ function.
+
+ .. warning::
+
+ This is not the denominator of the rational function
+ defined by self, which would always be 1 since self is a
+ polynomial.
+
+ EXAMPLES:
+
+ First we compute the denominator of a polynomial with
+ integer coefficients, which is of course 1.
+
+ ::
+
+ sage: R.<x,y> = ZZ[]
+ sage: f = x^3 + 17*y + x + y
+ sage: f.denominator()
+ 1
+
+ Next we compute the denominator of a polynomial over a number
field.
+
+ ::
+
+ sage: R.<x,y> = NumberField(symbolic_expression(x^2+3)
,'a')['x,y']
+ sage: f = (1/17)*x^19 + (1/6)*y - (2/3)*x + 1/3; f
+ 1/17*x^19 - 2/3*x + 1/6*y + 1/3
+ sage: f.denominator()
+ 102
+
+ Finally, we try to compute the denominator of a polynomial with
+ coefficients in the real numbers, which is a ring whose elements
do
+ not have a denominator method.
+
+ ::
+
+ sage: R.<a,b,c> = RR[]
+ sage: f = a + b + RR('0.3'); f
+ a + b + 0.300000000000000
+ sage: f.denominator()
+ 1.00000000000000
+ """
+ if self.degree() == -1:
+ return 1
+ #R was defined for univariate polynomials. Is it useless?
+ #R = self.base_ring()
+ x = self.coefficients()
+ try:
+ d = x[0].denominator()
+ for y in x:
+ d = d.lcm(y.denominator())
+ return d
+ except(AttributeError):
+ return self.parent().one_element()
+
+ def numerator(self):
+ """
+ Return a numerator of self computed as self * self.denominator()
+
+ Note that some subclases may implement its own numerator
+ function.
+
+ .. warning::
+
+ This is not the numerator of the rational function
+ defined by self, which would always be self since self is a
+ polynomial.
+
+ EXAMPLES:
+
+ First we compute the numerator of a polynomial with
+ integer coefficients, which is of course self.
+
+ ::
+
+ sage: R.<x, y> = ZZ[]
+ sage: f = x^3 + 17*x + y + 1
+ sage: f.numerator()
+ x^3 + 17*x + y + 1
+ sage: f == f.numerator()
+ True
+
+ Next we compute the numerator of a polynomial over a number
field.
+
+ ::
+
+ sage: R.<x,y> = NumberField(symbolic_expression(x^2+3)
,'a')['x,y']
+ sage: f = (1/17)*y^19 - (2/3)*x + 1/3; f
+ 1/17*y^19 - 2/3*x + 1/3
+ sage: f.numerator()
+ 3*y^19 - 34*x + 17
+ sage: f == f.numerator()
+ False
+
+ We try to compute the numerator of a polynomial with coefficients
in
+ the finite field of 3 elements.
+
+ ::
+
+ sage: K.<x,y,z> = GF(3)['x, y, z']
+ sage: f = 2*x*z + 2*z^2 + 2*y + 1; f
+ -x*z - z^2 - y + 1
+ sage: f.numerator()
+ -x*z - z^2 - y + 1
+
+ We check that the computation the numerator and denominator
+ are valid
+
+ ::
+
+ sage:
K=NumberField(symbolic_expression('x^3+2'),'a')['x']['s,t']
+ sage: f=K.random_element()
+ sage: f.numerator() / f.denominator() == f
+ True
+ sage: R=RR['x,y,z']
+ sage: f=R.random_element()
+ sage: f.numerator() / f.denominator() == f
+ True
+ """
+ return self * self.denominator()
+
cdef remove_from_tuple(e, int ind):
w = list(e)
del w[ind]
}}}
{{{
diff -r ca1f31d6f6bf
sage/rings/polynomial/multi_polynomial_libsingular.pyx
--- a/sage/rings/polynomial/multi_polynomial_libsingular.pyx Thu Jul 09
15:14:36 2009 -0700
+++ b/sage/rings/polynomial/multi_polynomial_libsingular.pyx Wed Jul 15
20:51:20 2009 -0700
@@ -4866,7 +4866,69 @@
i.append( new_MP(self._parent, pDiff(self._poly, k)))
return i
-
+
+ def numerator(self):
+ """
+ Return a numerator of self computed as self * self.denominator()
+
+ If the base_field of self is the Rational Field then the
+ numerator is a polynomial whose base_ring is the Integer Ring,
+ this is done for compatibility to the univariate case.
+
+ .. warning::
+
+ This is not the numerator of the rational function
+ defined by self, which would always be self since self is a
+ polynomial.
+
+ EXAMPLES:
+
+ First we compute the numerator of a polynomial with
+ integer coefficients, which is of course self.
+
+ ::
+
+ sage: R.<x, y> = ZZ[]
+ sage: f = x^3 + 17*y + 1
+ sage: f.numerator()
+ x^3 + 17*y + 1
+ sage: f == f.numerator()
+ True
+
+ Next we compute the numerator of a polynomial with rational
+ coefficients.
+
+ ::
+
+ sage: R.<x,y> = PolynomialRing(QQ)
+ sage: f = (1/17)*x^19 - (2/3)*y + 1/3; f
+ 1/17*x^19 - 2/3*y + 1/3
+ sage: f.numerator()
+ 3*x^19 - 34*y + 17
+ sage: f == f.numerator()
+ False
+ sage: f.numerator().base_ring()
+ Integer Ring
+
+ We check that the computation the numerator and denominator
+ are valid
+
+ ::
+
+ sage: K=QQ['x,y']
+ sage: f=K.random_element()
+ sage: f.numerator() / f.denominator() == f
+ True
+ """
+ if self.base_ring() == RationalField():
+ #This part is for compatibility with the univariate case,
+ #where the numerator of a polynomial over RationalField
+ #is a polynomial over IntegerRing
+ integer_polynomial_ring =
MPolynomialRing_libsingular(ZZ,self.parent().ngens(),self.parent().gens(),order=self.parent().term_order())
+ return integer_polynomial_ring(self * self.denominator())
+ else:
+ return self * self.denominator()
+
def unpickle_MPolynomial_libsingular(MPolynomialRing_libsingular R, d):
"""
Deserialize an ``MPolynomial_libsingular`` object
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6496#comment:2>
Sage <http://sagemath.org/>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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