On heights of polynomials with real roots. (Q2770822)

Let \(P\in Z[X]\), and assume that \(P\) has \(d\) real zeros and does not vanish at \(0,\pm1\). \textit{A. Schinzel} [Acta Arith. 24, 385--399 (1973; Zbl 0275.12004)] gave a lower bound for the Mahler measure \(M(P)\), depending on \(d\), and later other proofs of this result were found. The author gives two more proofs, one of which leads to a refined version of Schinzel's result. Moreover, an asymptotic result for the Remak height of Chebyshev polynomials is obtained. (The Remak height \(R(P)\) of a polynomial \(P(X)=\prod_{i=1}^n(X-\alpha_i)\) (with non-zero \(\alpha_i\)'s) is defined by NEWLINE\[NEWLINER(P)=\sqrt{| P(0)| }M(g)^{1/(2n-2)},NEWLINE\]NEWLINE where \(g(X)=\prod_{i\neq j}(X-\alpha_i/\alpha_j).\)