Smyth, C. J. [Myerson, Gerald] 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. Reviewer: D. R. Heath-Brown (Oxford) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 9 ReviewsCited in 44 Documents 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 Keywords:measure of polynomials; explicit evaluation; asymptotic values Citations:Zbl 0474.12005 PDFBibTeX XMLCite \textit{C. J. Smyth}, Bull. Aust. Math. Soc. 23, 49--63 (1981; Zbl 0442.10034) Full Text: DOI 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.