## 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.)

### Keywords:

polynomials; algebraic number; Mahler measure

### Citations:

Zbl 0788.14017; Zbl 0786.11063; Zbl 0861.14018
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