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The integer transfinite diameter of intervals and totally real algebraic integers. (English) Zbl 0892.11033
Given an interval $$I$$ of the real line, the integer transfinite diameter of $$I$$ is defined by $$t_{\mathbb Z}(I) = \inf_P \| P\| _\infty^{1/\partial P}$$, where $$\| P\| _\infty$$ is the maximum of $$| P(x)|$$ on $$I$$ and the infimum is over all non-constant $$P$$ with integer coefficients. As usual $$\partial P$$ denotes the degree of $$P$$. The classical transfinite diameter $$t(I)$$ has the same definition except that the infimum is over all polynomials with complex coefficients. It is known that $$t(I) = | I| /4$$ and for intervals of length greater than $$4$$ that $$t_{\mathbb Z}(I) = t(I)$$. On the other hand the exact value of $$t_{\mathbb Z}(I)$$ is not known for any interval of length less than $$4$$.
This interesting paper studies $$t_{\mathbb Z}(I)$$ for small intervals with one fixed rational endpoint. The authors define and study certain functions $$g_-,g, g_+$$ which give lower and upper bounds for $$t_{\mathbb Z}(I)$$ for Farey intervals $$[p/q,r/s]$$ with $$qr-ps = 1$$. These are used to prove good upper and lower bounds for $$t_{\mathbb Z}(I)$$ for intervals of the form $$[r/s,r/s + \delta]$$ or $$[r/s - \delta,r/s]$$, where $$r/s$$ is fixed and $$\delta \to 0$$. These results generalize those of F. Amoroso [Ann. Inst. Fourier 40, 885–911 (1990; Zbl 0713.41004)] and P. Borwein and T. Erdélyi [Math. Comput. 68, 661–681 (1996; Zbl 0859.11044)].
Extending results of the latter authors, they define critical polynomials $$P$$ and critical values $$c_P(I)$$ for an interval $$I$$ which have the property that if $$Q \in \mathbb Z[x]$$ has $$\| Q\| _\infty^{1/\partial Q} < c_P$$ then $$P^k$$ divides $$Q$$, where $$k \geq \gamma \partial Q$$, $$\gamma$$ being a constant depending only on $$P$$ and $$\| Q\| _\infty$$. They determine 10 critical polynomials for $$[0,1]$$, extending results of E. Aparicio [J. Approximation Theory 55, 270–278 (1988; Zbl 0663.41008)] and Borwein and Erdélyi (loc. cit.). Finally they obtain from their results an interesting result concerning totally real algebraic integers, namely that if such an $$\alpha$$ has least conjugate $$\alpha_1$$, then the mean value of $$\alpha$$, $$\text{Trace} (\alpha)/\partial \alpha > 1.6 + \alpha_1$$, with 8 explicitly listed exceptions.

##### MSC:
 11R04 Algebraic numbers; rings of algebraic integers 11R09 Polynomials (irreducibility, etc.) 11R80 Totally real fields
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##### References:
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