RE: Division of multi-var polynomials

*To*: mathgroup at yoda.physics.unc.edu*Subject*: RE: Division of multi-var polynomials*From*: fulling at sarastro.tamu.edu (Stephen A. Fulling)*Date*: Wed, 1 Apr 92 13:54:02 CST

>Date: Fri, 27 Mar 92 15:03:29 EST >To: mathgroup at yoda.physics.unc.edu >From: prm9c at hagar2.acc.virginia.edu (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.