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New methods providing high degree polynomials with small Mahler measure. (English) Zbl 1075.11065

Apparently, D.W. Boyd was the first who gave a list of polynomials of bounded degree and bounded Mahler measure. This computational work was then continued by M. J. Mossinghoff. Mossinghoff’s list contains all noncyclotomic and irreducible polynomials, with Mahler’s measure less than \(1.3\) and with degree at most \(180\). In this paper, the authors suggest two new methods of producing polynomials of high degree and of small Mahler measure. One of their methods is based on statistical considerations. By considering polynomials that in some sense are “close” to \(x^d+1\), they produce a list of polynomials up to degree \(300\). For instance, the Mahler measure of the polynomial \(x^{242}-x^{179}+x^{121}-x^{63}+1\) is equal to \(1.28571\dots\). Their second method is based on a certain minimization algorithm. One starts with an initial reciprocal polynomial and then, by changing a pair of reciprocal coefficients by \(1\), approaches to a local minima. Starting, for instance, with the polynomial \(x^{10}+x^9+x^8+x^7-3x^6-3x^5-3x^4+x^3+x^2+x+1\) with Mahler measure \(3.62614\dots\) they obtain (after four steps of their algorithm) the Lehmer polynomial \(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1\) with Mahler measure \(1.17628\dots\).

MSC:

11R09 Polynomials (irreducibility, etc.)
11Y40 Algebraic number theory computations
12-04 Software, source code, etc. for problems pertaining to field theory
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References:

[1] Boyd D. W., Math. Comp. 35 pp 1361– (1980)
[2] Boyd D. W., Canad. Math. Bull. 24 pp 453– (1981) · Zbl 0474.12005
[3] Boyd D. W., Math. Comp. 53 pp 355– (1989) · Zbl 0684.12003
[4] Cartea A., ”Probability and Distribution Theory.” (1999)
[5] Mossinghoff M. J., Math. Comp. 67 pp 1697– (1998) · Zbl 0918.11056
[6] Mossinghoff M. J., ”Small Salem Numbers.” (2002)
[7] Smyth C. J., Bull. London. Math. Soc. 3 pp 169– (1971) · Zbl 0235.12003
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