## 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 $$\equiv 1\pmod n$$.

### MSC:

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

### Keywords:

Lehmer’s problem; Schinzel and Zassenhaus conjecture
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