# [Maxima] determine if expression is polynomial P(x,y/Q(x,y)

Stavros Macrakis macrakis at alum.mit.edu
Sat Feb 25 12:41:50 CST 2012

```You can often tell if something can be written as a polynomial by seeing if
its Taylor series "runs out", e.g.

taylor(cos(6*acos(x)),x,0,*100*) => -1+18*x^2-48*x^4+32*x^*6* +...

This requires only arithmetic -- no clever simplifications, no
pattern-matching, etc.  Of course, this doesn't *guarantee* that the 101+th
term is zero, but in many 'natural' expressions, it has to be.

I wonder if there is some sort of analog to this for univariate rational
functions?

-s

On Sat, Feb 25, 2012 at 12:47, Richard Fateman <fateman at eecs.berkeley.edu>wrote:

>  On 2/25/2012 9:34 AM, Stavros Macrakis wrote:
>
>   But beware of things like trigonometric identities (as RJF says),
>
>                -s
>
>  One of my favorite polynomial-producing expressions is along the lines
> of cos(p*acos(x)).
>
> For example,  trigexpand(cos(5*acos(x))  is
> 5*x*(1-x^2)^2+x^5-10*x^3*(1-x^2) = 16*x^5-20*x^3+5*x
>
>
>
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