×

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

Citations:

Zbl 0474.12005
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bhattacharya, Normal approximations and asymptotic expansions (1976)
[2] Smyth, Bull. Austral. Math. Soc. 23 pp 49– (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.