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.


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
Full Text: DOI


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