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An upper bound for the length of cyclotomic polynomials. (Une majoration de la longueur des polynômes cyclotomiques.) (French) Zbl 0992.11028
The authors improve a result of Bateman on the length \(\beta(m)\) of the \(m\)-th cyclotomic polynomial \(\Phi_m\), i.e. the sum of the absolute values of its coefficients. P. T. Bateman [Bull. Am. Math. Soc. 55, 1180-1181 (1949; Zbl 0035.31102)] proved the upper estimate \(\beta(m)\leq m^{\frac 12 d(m)}\) where \(d\) denotes the number of divisors of \(m\). P. T. Bateman, C. Pomerance and R. C. Vaughan [Topics in Classical Number Theory, Colloq. Math. János Bolyai 34, 171-202 (1984; Zbl 0547.10010)] proved that \(\beta(m)\) can take large values for some \(m\) whereas for small values this phenomenon does not exist. In this paper the authors prove \(\beta(m)< (\sqrt{2})^{\varphi(m)}\) for \(m\geq 7\) and \(m\neq 10\), where \(\varphi\) denotes Euler’s function.
From this result they deduce for the polynomial \(P_m(X)= \Phi_m(X)- (X-1)^{\varphi(m)}\) for \(m\geq 2\) that if this polynomial vanishes at some root of unity, then this root of unity is of order 6.
This is related to a conjecture of the second author [Acta Arith. 54, 87-125 (1989; Zbl 0642.12010)] and a result of C. Hélou on Cauchy-Mirimanoff polynomials [see C. R. Math. Acad. Sci., Soc. R. Can. 19, 51-57 (1997; Zbl 0897.11032)]. Moreover, the authors have extended Terjanian’s conjecture up to \(m=1000\) using Maple and give a description of their method.

MSC:
11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)
11T22 Cyclotomy
Software:
Maple
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