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The mean values of totally real algebraic integers. (English) Zbl 0536.12006

Let \(\alpha\) be a totally real algebraic integer of degree \(d\) with conjugates \(\alpha_1,\ldots,\alpha_d\). Let \(p>0\) and then define \(M_p(\alpha)=(d^{-1}\sum^d_{i=1}| \alpha_i|^d)^{1/p}\). The author investigates the structure of the set \(\mathcal M_p\) of values of \(M_p(\alpha)\) as \(\alpha\) varies over all totally real algebraic integers of all degrees. For each \(p>0\), he determines two numbers \(A_p<B_p\) so that \(\mathcal M_p\cap(1,A_p)\) is a finite non-empty set whose members are given explicitly, and such that \(\mathcal M_p\) is dense in \((B_p,\infty)\). Thus the smallest element of \(\mathcal M_p\) is known for all \(p>0\). The author conjectures that \(B_p\) is, in fact, the smallest limit point of \(\mathcal M_p\).
The structure of \(\mathcal M_p\) is thus similar to the set of values of \(\Omega(\alpha)=\prod \max(1,| \alpha_ i|)^{1/d}\) which he considered in two earlier papers [J. Aust. Math. Soc., Ser. A 30, 137–149 (1980; Zbl 0457.12001); Math. Comput. 37, 205–208 (1981; Zbl 0475.12001)]. The results improve upon and supersede earlier results of C. L. Siegel \((p=2)\) [Ann. Math. (2) 46, 302–312 (1945; Zbl 0063.07009)], J. Hunter \((p=4)\) [Proc. Glasg. Math. Assoc. 2, 149–158 (1956; Zbl 0074.03901)] and M. J. McAuley \((p=1,2,3,4,6,8,10,12)\) [M. Sc. thesis, Univ. Adelaide, 1981].
The proofs are a nice blend of computation and analysis.

MSC:

11R80 Totally real fields
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R04 Algebraic numbers; rings of algebraic integers
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