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

Synopsis

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:

For the programmer

The object diffRatFun is a method function.