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
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.)
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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.
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.
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