Re: quasi-cyclotomic polynomials
- From: Brian VanPelt <brvanpelt@xxxxxxxxxxxxx>
- Date: Fri, 24 Aug 2007 22:19:32 -0400
On Fri, 24 Aug 2007 20:04:28 -0400, quasi <quasi@xxxxxxxx> wrote:
On Fri, 24 Aug 2007 19:59:01 -0400, Brian VanPelt
<brvanpelt@xxxxxxxxxxxxx> wrote:
On Fri, 24 Aug 2007 19:35:49 -0400, quasi <quasi@xxxxxxxx> wrote:
The ideas below were inspired by a question asked by tommy1729.
Thanks again, tommy.
Some of tommy's "crazy" questions are crazy-cool, while others are
just plain crazy. But even the truly crazy ones seem to have
intriguing offshoots.
Ok, the ideas and questions ...
For this discussion, all polynomials will be elements of Z[x].
Call a polynomial "quasi-cyclotomic" if all of its roots have complex
modulus equal to 1.
Note -- my use of "quasi" in this context is not in any way connected
with my username. Rather, it's because the above definition apparently
generalizes the notion of cyclotomic polynomials.
Perhaps these polynomials already have a name? If so, what name?
A few simple observations ...
(1) All nonzero constant polynomials are automatically
quasi-cyclotomic, since they have no roots.
(2) Every cyclotomic polynomial is quasi-cyclotomic.
(3) The set of quasi-cyclotomic polynomials is a saturated,
multiplicatively closed subset of Z[x].
A few questions ...
Question (1):
Does there exist a nonconstant, primitive, quasi-cyclotomic polynomial
whose leading coefficient is not 1 or -1?
Note -- by primitive, we mean that the coefficients have no common
factor (in Z).
Question (2):
Does there exist a nonconstant, monic, irreducible quasi-cyclotomic
polynomial which is not cyclotomic?
Note -- if the answer to question (2) is "no", then the answer to
question (1) is automatically also "no".
quasi
Check out this reference
http://mathworld.wolfram.com/CyclotomicPolynomial.html
In particular, look at (44) on.
Nice reference, but I don't see how (44) answers either question.
quasi
I'm sorry, I wasn't trying to give a solution. I thought you were
interested in some research in that area. I was looking at (44) on,
and the links at the end of the article. There has been a lot of
research in this area - my link was simply meant to get you started.
Brian
.
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