Upper bounds for the coefficients of irreducible integer polynomials in several variables (Q2717762)

M. Mignotte and later F. Amoroso, M. Waldschmidt, Y. Bugeaud, the reviewer and others obtained various estimates from above for the height of a polynomial in terms of its degree and the Mahler measure which are stronger than the classical Liouville-type bound provided that the Mahler measure is small. In this paper, these results are extended to polynomials in several variables. The main result (Theorem 1) is a certain inequality which leaves one parameter at the disposition of the authors. By choosing the parameter in several ways, one can obtain some interesting and simply looking corollaries. NEWLINENEWLINENEWLINELet \(M(P)\) be the Mahler measure of an irreducible polynomial \(P\) in \(n\) variables with integer coefficients of degree \(D\), and let \(|P|\) be the maximum modulus of \(P\) over the polydisk \(|z_1|=\dots=|z_n|=1\). For instance, by Corollary 1 of the paper, we have that NEWLINE\[NEWLINE \log |P|< 2D^{n/(n+1)} \log (D^3 (n+2)^3 M(P)). NEWLINE\]NEWLINE This improves (essentially by \(D^{1/(n+1)}\) in the exponent) the Liouville-type inequality \(|P|\leq (n+1)^D M(P)\), if say \(D\) is large compared to \(n\) and \(M(P)\). It follows that, given an irreducible integer polynomial \(P\) of partial degrees \(D_1, \dots, D_n\) and algebraic numbers \(\alpha_1, \dots, \alpha_n\), we have either \(P(\alpha_1, \dots, \alpha_n)=0\) or NEWLINE\[NEWLINE |P(\alpha_1, \dots, \alpha_n)|\geq (D^3 (n+2)^3 M(P))^{-2(d-1)D^{n/(n+1)}} M(\alpha_1)^{-D_1d/d_1} \dots M(\alpha_n)^{-D_nd/d_n}, NEWLINE\]NEWLINE where \(d=[{\mathbb{Q}}(\alpha_1, \dots, \alpha_n): {\mathbb{Q}}]\) and \(d_j\) is the degree of \(\alpha_j\) over \(\mathbb{Q}\). For \(M(P)<e^{D/4-1}\), Corollary 4 implies that NEWLINE\[NEWLINE \log |P|< (n+1) D^{n/(n+1)} (\log (e M(P)))^{1/(n+1)} \log (4D^2 (n+2)). NEWLINE\]NEWLINE The latter inequality is shown to be sharp up to a power of \(\log D\) in the narrow range NEWLINE\[NEWLINE D (\log (D(n+2)))^{-c}\ll \log M(P) \ll D \log (D(n+2)), NEWLINE\]NEWLINE where \(c>0\) is fixed.