Lehmer’s problem for polynomials with odd coefficients. (English) Zbl 1172.11034

Let \(f\in\mathbb Z[x]\) have odd coefficients, degree \(n-1\) and no cyclotomic factors. Then the authors show that the Mahler measure \(M(f)\) of \(f\) satisfies \[ \log M(f)\geq \frac{\log 5}{4}\left(1-\frac{1}{n}\right). \tag{*} \] This solves the well-known problem of D.H. Lehmer in the case of irreducible polynomials with odd coefficients. The result is generalised to polynomials \(f\) (degree \(n-1\), no cyclotomic factors) whose coefficients are \(\equiv 1\pmod{n}\): in this case \(\log M(f)\geq \log(m/2)(1-1/n)\). These results come from a more general theorem (Theorem 3.3) involving an auxiliary function \(F\). In the case of such \(f\) having odd coefficients (\(m=2\)) this states that if \(F\) has integer coefficients and \(\gcd(f(x),F(x^n))=1\) then \[ \log M(f)\geq \frac{\nu(F)\log 2-\log\|F\|_\infty}{\deg F}\left(1-\frac{1}{n}\right). \] Here \(\|F\|_\infty=\max_{|z|=1}|F(z)|\) and \(\nu(F)\) is the sum of the multiplicities of all factors of \(F\) that are \(2^k\)th cyclotomic polynomials for some \(k\). The corollary (*) is obtained by choosing \(F(x)=(1+x^2)(1-x^2)^4\), the condition that \(f\) has no cyclotomic factors being required to ensure that \(\gcd(f(x),F(x^n))=1\). The proof makes use of upper and lower bounds for the (nonzero) resultant \(\text{Res}(f(x),F(x^n))\).
Also, the conjecture of Schinzel and Zassenhaus is resolved for the case of polynomials \(f\) with odd coefficients (degree \(n-1\), at least one noncyclotomic factor). The authors show that such an \(f\) has a root of modulus greater than \(1+\log 3/n\). They also give the lower bound \(1+\log(m-1)/n\) for the largest root of \(f\) when \(m>2\) and the coefficients of \(f\) are all \(\equiv 1\pmod n\).


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11C08 Polynomials in number theory
11R09 Polynomials (irreducibility, etc.)
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