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RE: Division of multi-var polynomials

  • To: mathgroup at
  • Subject: RE: Division of multi-var polynomials
  • From: fulling at (Stephen A. Fulling)
  • Date: Wed, 1 Apr 92 13:54:02 CST

>Date: Fri, 27 Mar 92 15:03:29 EST
>To: mathgroup at
>From: prm9c at (Peter Richard Molnar)
>Subject: Division of multi-var polynomials
>       I am trying to divide one polynomial into another, each of
>which has three variables, to get a series of terms.
>For example,
>X^2 * Y^3 * W/(1-2*X*Y*W) = X^2*Y^3*W + 2*X^3*Y^4*W^2 + 4*X^4*Y^5*W^3
+ ...
>The above result will be an infinite series of terms, but I
>would like mathmatica to give me the first N terms.  Mathmatica
>doesn't seem to be able to handle more than one variable when
>dividing polynomials, nor does it seem to be able to do the
>above manner of division.  If anyone can tell me how to do this
>on Mathmatica it would be greatly appreciated.

      What you are doing is replacing the function 1/(1-z) by its
Taylor series, 1 + z + z^2 + ..., and multiplying by the numerator.
In this example z = 2*X*Y*W, but z could be anything (such as a
polynomial of more than one term).  This can be done within Mma
either "by hand" or using Mma's functions for power series in one
variable.  (If the constant term in the denominator is not 1, divide
out by it first.  If there is no constant term, as in (X*Z)/(X+Y+Y^2),
then your function just doesn't have a polynomial approximation
about the origin (unless there is a cancellation).)

      I'm not sure what "the first N terms" means for a series in
more than one variable.  It is easy enough to truncate the series in
z after z^N, but probably what you want is to automatically throw away
all terms in which the sum of the exponents of X, Y, and W is greater
than some M, say M = 4.  One drastic way to do this is to unprotect
and redefine Times[] so that X^5 := 0, X^4*Y := 0, etc.  Then remember
to reset everything before you have a chance to use multiplication in
another context!  There should be a better way to do this.  The Mma
book contains only one example of a series in two variables (p. 429
of 1st edition = p. 641 of 2nd edition), and it treats the variables
hierarchically, not as a true multivariate power series.

					 S. A. Fulling
					 Texas A&M Math.

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