Let \(f(x)\) be a nonzero polynomial with complex coefficients and degree at most \(D\) and let \(H(f)\), \(L(f)\), \(M(f)\), \(\|f\|= \max_{|z|=1|}|f(z)|\) denote the usual height, length, Mahler measure and norm of \(f\). Let \(r_\rho(f)\) denote the order of vanishing of \(f\) at a complex number \(\rho\). For a given set \(\mathcal A\) of complex numbers, let \(r_\rho(\mathcal A,D)\) be the maximum of \(r_\rho(f)\) over all non-zero \(f\) of degree \(D\) with coefficients in \(\mathcal A\).
Theorem 1 shows that \(r_{-1}(\{0,1\},D)\) is one of the two values \([\log D/\log 2]\) or \([\log D/\log 2] - 1\) and that each possibility occurs infinitely often. This refines a result of P. Borwein, T. Erdélyi and G. Kós [Proc. Lond. Math. Soc. 79, 22-46 (1999; Zbl 1039.11046)]. The author considers the question of bounding \(|P(1)|\) in terms of \(M\) for irreducible polynomials of degree \(d\) with \(M(P) \leq M\). He constructs a sequence of irreducible polynomials \(Q_d(x)\) of degree \(d\) such that \(Q_d(1) > \exp((1-\varepsilon)\sqrt{d \log d})\), for sufficiently large \(d\). If \(\alpha\) is a root of \(Q_d\) then \(\mathcal N(\alpha-1) > \exp((1-2\varepsilon)\sqrt{d \log M(\alpha)})\), where \(\mathcal N\) denotes the norm function. He shows how this gives an improvement of Schur’s bound for \(s(L,D)\), the maximum of \(r_1(f)\) over all polynomials of degree \(D\) and length at most \(L\).
Finally, he considers the question of bounding \(\min |\alpha+1|\) over all roots \(\alpha \neq -1\) of polynomials of degree \(D\) with coefficients in the set \(\mathcal A\). He obtains improvements on all the known upper and lower bounds for the three cases \(\mathcal A = \{-1,0,1\}, \{-1,1\}\) and \(\{0,1\}\). The relationship to previous work is carefully described.