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This is inspired by this question. Suppose I give you an integer $k.$ Is there a classification of cyclotomic polynomials with exactly $k$ nonzero terms? For example, if $k=1$ there are not any, if $k=2,$ you get just the standard $x^n-1$ polynomials, if $k=3,$ I assume you only get the squares of the standard polynomials, for $k=4$ presumably you only get the products of distinct standard cyclotomics, but after that it becomes a bit puzzling...

Edit by cyclotomic polynomial I mean one all of whose roots are roots of unity.

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When you say cyclotomic polynomial do you just mean a monic integer polynomial all of whose roots are on the unit circle? That is not the usual definition: the minimal polynomial of a root of unity $e^{\frac{2i\pi}{n}}$. It is a non-trivial fact that the first definition describes exactly the products of cyclotomic polynomials. – Aaron Meyerowitz May 26 '12 at 6:23
@Aaron, yes, that is what I mean. I will edit the question to clarify, thanks!. – Igor Rivin May 26 '12 at 13:56

For $k=2$, you get $t^{2^n}+1 = C_{2^{n+1}}(t)$. For $k=3$, see For $k=5$, see See also

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Some of the following is mentioned in the OEIS references given by Robert:

Let $a_n$ be the number of terms in $\phi_n(t)$ , the $n$th cyclotomic polynomial. Then $a_n=a_m$ where $m$ is the product of the prime divisors of $n$ because $$\phi_n(t)=\phi_m(t^{n/m}).$$ So we only need to figure out $a_m$ for $m$ square-free.

For odd $n$ $a_{2n}=a_n.$

For odd prime $p$, $a_p=p$ and, if $q \ne p$ is prime, then $$a_{pq}=\frac{2(p-u)(uq+1)}{p}-1$$ where $1 \le u \le p-1$ and $uq \equiv -1 \mod p.$ This should allow one to determine all cases where $a_n=k$ and $n$ has at most two distinct odd prime factors. Quite a bit more can be said in this special case, the non-zero coefficients are all $\pm 1$ and one can say exactly where they appear.

I'm not sure just how much is known about $a_{pqr}$ for $p,q,r$ distinct odd primes. Perhaps lower bounds can be given. Here are a few cases: $[105, 33], [165, 57], [195, 59], [231, 57], [255, 73], [273, 99], [357, 125], [385, 177], [429, 135]$$ [455, 189], [561, 107], [595, 253], [663, 129], [715, 213], [935, 429], [1001, 321]$ So it would seem that perhaps for up to $30$ terms or so everything can be said, but that is just a guess on my part.

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I may be misreading things, but I believe you Aaron and Robert are talking about minimal polynomials which are often called (by me and others) cyclotomic polynomials, while Igor unforunately is using the term for what we would call products of cyclotomic polynomials. Although your a_n are of interest, they do not address the question I see as being asked. In particular, I think Igor is asking about a class that includes x+1 times any cyclotomic polynomial. Gerhard "Ask Me About System Design" Paseman, 2012.05.26 – Gerhard Paseman May 26 '12 at 16:05

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