## 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.)

### Keywords:

Mahler measure; conjecture of Bogomolov

### Citations:

Zbl 0788.14017; Zbl 0786.11063
Full Text:

### References:

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