classification of cyclotomic fewnomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:22:03Z http://mathoverflow.net/feeds/question/97989 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97989/classification-of-cyclotomic-fewnomials classification of cyclotomic fewnomials Igor Rivin 2012-05-25T21:32:43Z 2012-05-26T14:46:39Z <p>This is inspired by <a href="http://mathoverflow.net/questions/97952/when-is-xyz-1-x-1-y-1-z-cyclotomic" rel="nofollow">this question.</a> 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...</p> <p><strong>Edit</strong> by cyclotomic polynomial I mean one all of whose roots are roots of unity.</p> http://mathoverflow.net/questions/97989/classification-of-cyclotomic-fewnomials/97997#97997 Answer by Robert Israel for classification of cyclotomic fewnomials Robert Israel 2012-05-25T22:14:01Z 2012-05-25T23:23:53Z <p>For $k=2$, you get $t^{2^n}+1 = C_{2^{n+1}}(t)$. For $k=3$, see <a href="http://oeis.org/A065119" rel="nofollow">http://oeis.org/A065119</a>. For $k=5$, see <a href="http://oeis.org/A086761" rel="nofollow">http://oeis.org/A086761</a> See also <a href="http://oeis.org/A051664" rel="nofollow">http://oeis.org/A051664</a></p> http://mathoverflow.net/questions/97989/classification-of-cyclotomic-fewnomials/98018#98018 Answer by Aaron Meyerowitz for classification of cyclotomic fewnomials Aaron Meyerowitz 2012-05-26T06:10:08Z 2012-05-26T06:10:08Z <p>Some of the following is mentioned in the OEIS references given by Robert:</p> <p>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.</p> <p>For odd $n$ $a_{2n}=a_n.$</p> <p>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. <a href="http://www.jstor.org.ezproxy.fau.edu/stable/2314500" rel="nofollow">Quite a bit more</a> can be said in this special case, the non-zero coefficients are all $\pm 1$ and one can say exactly where they appear.</p> <p>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.</p>