## 2009年12月28日 星期一

### Polynomials and topology

Recently my friend is working on a problem in topology, and out of his work, it appears that the there is a special pattern in the coefficients of the following polynomial, when m,n are relatively prime:

$\frac{(x^{mn}-1)(x-1)}{(x^m-1)(x^n-1)} (x^{4n}-x^{2n}+1)$

It appears that if one expands this polynomial out, collects terms and arranges them in decreasing powers of x, then the non-zero coefficients are all either 1 and -1, and they appear to alternate as the power decreases. (e.g. when m=4, n=3, the polynomial is

$x^{18}-x^{17}+x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x+1$

It is not known whether this pattern really exists. But I thought this is cute and may be of interest to some of you. Does any of you have any idea about how to prove/disprove it?

(The case of interest in topology is when m > 3n, but it looks like this pattern persists as long as m,n are relatively prime.)

#### 2 則留言:

Kahoo 提到...

I am not sure if the pattern really exists, but I recall seeing something similar when I learned about "cyclotomic polynomials" -- the coefficients also appear to be 1, 0 or -1 all the time when the power is small, and it was not until power 105 when the first counterexample can be found.

Polam 提到...

Thanks Kahoo! We thought a little about cyclotomic polynomials, and it was helpful that you mentioned the power 105 in connection to that. We still don't know though whether there is a counterexample.