## On measures of polynomials in several variables.(English)Zbl 0442.10034

Bull. Aust. Math. Soc. 23, 49-63 (1981); corrigendum 26, 317-319 (1982).
The measure of a non-zero polynomial $$P\in\mathbb C[x_1,\dots, x_n]$$ was defined by Mahler to be $M(P)=\exp\left\{(2\pi)^n\int_0^{2\pi}\cdots\int_0^{2\pi} \log\left| P\left(e^{i\theta_1},\dots,e^{i\theta_n}\right)\right| d\theta_1\cdots d\theta_n\right\}.$
Mahler merely used $$M(P)$$ as a more suitable indication of the “size” of $$P$$ than the usual height. Following work of D. W. Boyd [Can. Math. Bull. 24, 453–469 (1981; Zbl 0474.12005)], the present paper explores $$M(P)$$ further. A simple formula is given covering certain cases in which $$P$$ factorises completely into linear factors. Other examples include $$P(x,y)=T_n(x+y)+l$$, where $$l=0, \pm 1$$ or $$\pm 2$$ and $$T_n$$ is a Chebyshev polynomial $$(T_n(2\cos \theta)=2\cos n\theta)$$. For $$n=2$$ one obtains $M(x^2+2xy+y^2)=2^{\tfrac12}\exp(2\beta/\pi),$ where $$\beta$$is the Catalan constant, i.e. $$\beta=1-3^{-2}+5^{-2}-7^{-2}+9^{-2}\dots\,$$. Asymptotic formulas are also given. One example is $M(x_0 + x_1+ \dots + x_n)=c\sqrt n+O(1),$ where $$c=1.11593\dots$$ is given explicitly (see the corrigendum below). The explicit formulae use Jensen’s theorem on zeros of an analytic function, whilst the asymptotic expressions are proved using a quantitative version of the central limit theorem.
Added in 1983: From the text of the corrigendum by G. Myerson and the author: In the paper reviewed, it was asserted in Theorem 3 that the measure $$M(x_0 +x_1+\dots + x_n)$$ is asymptotically $$c\sqrt n+O(1)$$, where $$c$$ is an explicit constant. The value of $$c$$ given was incorrect, and should be $$e^{\tfrac12 \gamma}$$ where $$\gamma$$ is Euler’s constant. This was pointed out by the first author. In fact $M(x_0 +x_1+ \dots + x_n)=e^{\tfrac12 \gamma}\sqrt n + O(\log n/\sqrt n), \tag{1}$ where we have tried to make amends by improving the error term.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11C08 Polynomials in number theory 12D05 Polynomials in real and complex fields: factorization

Zbl 0474.12005
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### References:

 [1] Bhattacharya, Normal approximations and asymptotic expansions (1976) [2] Smyth, Bull. Austral. Math. Soc. 23 pp 49– (1981)
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