diffRatFun -- derivative of a rational function in a Weyl algebra

Synopsis

Usage:

diffRatFun(m,f)

diffRatFun(m,g,f,a)
Inputs:
- f, a ring element, polynomial in a Weyl algebra $D$ in $n$ variables or ratinoal function in the fraction field of a polynomial ring in $n$ variables
- g, a ring element, polynomial in a Weyl algebra $D$ in $n$ variables
- m, a list, of nonnegative integers $m = \{m_1,...,m_n\}$
- a, an integer, an integer
Outputs:
- a ring element, the result of applying the product of the $(dx_i)^{m_i}$ to $f$
- a list, the result of applying the product of the $(dx_i)^{m_i}$ to $g/f^a$, written as (numerator,denominator,power of denominator)

Description

Let $D$ be a Weyl algebra in the variables $x_1,..x_n$ and partials $dx_1,..,dx_n$. Let $f$ be either a polynomial or rational function in the $x_i$ and $m = (m_1,..,m_n)$ a list of nonnegative integers. The function $f$ may be given as an element of a polynomial ring in the $x_i$ or of the fraction field of that polynomial ring or of $D$. This method applies the product of the $dx_i^{m_i}$ to $f$. In the case of the input $(m,g,f,a)$, where $f \neq 0$ and $g$ are both polynomials and $a$ is a nonnegative integer, it applies the product of the $dx_i^{m_i}$ to $g/f^a$ and returns the resulting derivative as (numerator,denominator,power of denominator), not necessarily in lowest terms.

i1 : QQ[x,y,z]

o1 = QQ[x..z]

o1 : PolynomialRing

i2 : m = {1,1,0}

o2 = {1, 1, 0}

o2 : List

i3 : f = x^2*y+z^5

      5    2
o3 = z  + x y

o3 : QQ[x..z]

i4 : diffRatFun(m,f)

o4 = 2x

o4 : QQ[x..z]

i5 : makeWA(QQ[x,y,z])

o5 = QQ[x..z, dx, dy, dz]

o5 : PolynomialRing, 3 differential variable(s)

i6 : m = {1,1,0}

o6 = {1, 1, 0}

o6 : List

i7 : f = x^2*y+z^5

      5    2
o7 = z  + x y

o7 : QQ[x..z, dx, dy, dz]

i8 : diffRatFun(m,f)

o8 = 2x

o8 : QQ[x..z, dx, dy, dz]

i9 : frac(QQ[x,y])

o9 = frac(QQ[x..y])

o9 : FractionField

i10 : m = {1,2}

o10 = {1, 2}

o10 : List

i11 : f = x/y

      x
o11 = -
      y

o11 : frac(QQ[x..y])

i12 : diffRatFun(m,f)

       2
o12 = --
       3
      y

o12 : frac(QQ[x..y])

i13 : makeWA(QQ[x,y,z])

o13 = QQ[x..z, dx, dy, dz]

o13 : PolynomialRing, 3 differential variable(s)

i14 : m = {1,2,1}

o14 = {1, 2, 1}

o14 : List

i15 : g = z

o15 = z

o15 : QQ[x..z, dx, dy, dz]

i16 : f = x*y

o16 = x*y

o16 : QQ[x..z, dx, dy, dz]

i17 : a = 3

o17 = 3

i18 : diffRatFun(m,g,f,a)

           3 2
o18 = (-36x y , x*y, 7)

o18 : Sequence

Caveat

Must be over a ring of characteristic $0$.

Ways to use diffRatFun :

diffRatFun(List,RingElement)
diffRatFun(List,RingElement,RingElement,ZZ)

For the programmer

The object diffRatFun is a method function.