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On the spectrum of the Zhang-Zagier height. (English) Zbl 0960.11047

For an algebraic number \(\alpha\) of degree \(d(\alpha)\) and Mahler’s measure \(\text{M}(\alpha)\), define \[ {\mathfrak M}(\alpha)= M(\alpha)^{1/d(\alpha)} \quad\text{and}\quad {\mathfrak H}(\alpha)= {\mathfrak M}(\alpha){\mathfrak M}(1-\alpha). \] According to Zhang and Zagier [S. Zhang, Ann. Math., II. Ser. 136, No. 3, 569-587 (1992; Zbl 0788.14017), D. Zagier, Math. Comput. 61, 485-491 (1993; Zbl 0786.11063)], if \(\alpha\) is not a root of the polynomial \(P(z)=z(z-1)(z^{2}-z+1)\), then \[ {\mathfrak H}(\alpha)\geq \sqrt{(1+\sqrt{5})/2}=1.272\dots\;. \] Denote by \(\Phi_{10}\) the \(10\)-th cyclotomic polynomial. The author proves that if \(\alpha\) is not a root of the polynomial \(P(z)\Phi_{10}(z)\Phi_{10}(1-z)\), then \({\mathfrak H}(\alpha)\geq 1.281\dots \). Also he shows that the smallest limit point \(x\) of the set \(\{{\mathfrak H}(\alpha)\), \(\alpha\) algebraic} lies in the range \(1.281\dots\leq x\leq 1.291\dots \). The author concludes his paper with interesting speculations.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
11G50 Heights
11R09 Polynomials (irreducibility, etc.)
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