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## Height of cyclotomic polynomials

Recall that the cyclotomic polynomial of order $n$ is $$\Phi_n(X)=\prod_{gcd(k,n)=1}(X-e^{2ik\pi/n}).$$ Its degree is $\phi(n)$ (Euler's indicator). Inversion of the Moebius formula $$X^n-1=\prod_{d|n}\Phi_d(X)^{\mu(n/d)}$$ implies that $\Phi_n\in{\mathbb Z}[X]$. The height of $\Phi_N$ is the maximal modulus of its coefficients. Thus the height of $$\Phi_p(X)=X^{p-1}+\cdots+X+1$$ ($p$ odd prime) is $1$. A cyclotomic polynomial is flat if its height is $1$. We know that the height of $\Phi_{2n}$ equals that of of $\Phi_n$, and also that if a prime $p$ divides $n$, then the height of $\Phi_{pn}$ equals that of $\Phi_n$. Therefore it is enough to analyse the case where $n=p_1\cdots p_\ell$ is the product of distinct odd primes. When $\ell=2$ it is known that $\Phi_{p_1p_2}$ is flat. It is known that there are infinitely many flat $\Phi_{p_1p_2p_3}$, but $\Phi_{105}$ ($105=3\cdot 5\cdot 7$) is not flat (its height equals $2$).

Question. What is known about the growth of the height of $\Phi_n$ ? Is there a bound of the form $C_\ell$ (thus extending $C_1=C_2=1$) ?

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http://oeis.org/classic/A013594 gives the smallest cyclotomic polynomial with a given coefficient, and the paper http://www.ams.org/mathscinet-getitem?mr=0364141 by Vaughan gives lower bounds for the growth of the max coefficient of the form exp(log 2 log n/log log n)

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 Vaughan improves the previous results by factor $\log2$ (under exponential). These do not refer to bounds for particular $C_\ell$'s but it's reasonable to compare his results (using a completely different analytic method) with the ones coming in the studies of the ternary case. – Wadim Zudilin Oct 15 2010 at 6:02

The problem of $C_3$ is discussed in the MPIM preprint "The family of ternary cyclotomic polynomials with one free prime" by Yves Gallot, Pieter Moree, and Robert Wilms. As far as I know Pieter works on the problem for several years; he once said that most of the conjectures in this area are false.

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 From this paper, it seems that there is no finite bound $C_\ell$. Instead, even if $\ell=3$, one has a bound in terms of $p_1$ ($p$ in the literature). Correct ? – Denis Serre Oct 13 2010 at 11:54 Yes, this is correct. – Wadim Zudilin Oct 13 2010 at 12:04

In this book, a result due to Beiter is mentioned (see Thm. 2.7), according to which the height $C_3$ is bounded by $p-k$ or $p-k-1$ according as $p = 4k+1$ or $p = 4k+3$, where the three primes are $p$, $q$ and $r$ with $p < q < r$. Conjecturally, the best possible bound in this case is $C_3 = (p+1)/2$.

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 Franz, Beiter's original conjecture was recently corrected in arXiv:0910.2770. – Wadim Zudilin Oct 13 2010 at 10:18 The value $(p+1)/2$ is conjectured by Sister Marion Beiter (AMM 75 (1968)), who proved it when either $q$ or $r$ is $kp±1$. According to E. Leher's thesis, this conjecture is false. I expected that an accurate proof had been found since then. – Denis Serre Oct 13 2010 at 11:52