On the Mahler measure of the composition of two polynomials. (English) Zbl 0868.11050

For \(P(x), T(x)\in \mathbb{Z} [x ]\), \(P\) irreducible of degree at least 2, \(T\) divisible by \(x\) but not an integral multiple of a power of \(x\), we prove that the absolute Mahler measure of \(P(T (x))\) is bounded below by a constant \(c_T> 1\), independent of \(P\). If \(P(T (x))\) is irreducible, this constant can be taken to be \(1+ {1\over {2(t- t_0+ 4t|T|)}}\), where \(|T|\) is the sum of the absolute values of the coefficients of \(T\), and \(x^{t_0}\) is the highest power of \(x\) dividing \(T\). A more complicated bound holds in the general case. The results generalise earlier work of S. Zhang and D. Zagier for the case \(T(x)= x^2- x\). The techniques of the proof are based on recent work of F. Beukers and D. Zagier on heights of points lying on one hypersurface but not on another.


11R09 Polynomials (irreducibility, etc.)
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