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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.

MSC:

11R09 Polynomials (irreducibility, etc.)
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