## 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

### Citations:

Zbl 0713.41004; Zbl 0859.11044; Zbl 0663.41008
Full Text:

### References:

  Amoroso, F., Sur le diamètre transfini entier d’un intervalle réel, Ann. Inst. Fourier Grenoble40 (1990,), 885-911. · Zbl 0713.41004  Aparicio, E., Neuvas acotaciones para la desviación diofántica uniforme minima a cero en [0, 1] y [0,1/4], VIJornadas de Matemáticas Hispano-Lusas, Santander (1979), 289-291.  Aparicio, E., Sobre unos sistemas de numeros enteros algebraicos de D.S. Gorshkov y sus aplicaciones al cálculo, Rev. Mat. Hisp.-Amer.41 (1981), 3-17.  Aparicio, E., On the asymptotic structure of the polynomials of minimal Diophantic deviation from zero, J. Approx. Th.55 (1988), 270-278. · Zbl 0663.41008  Borwein, P. and Erdélyi, T., The integer Chebyshev problem, Math. Comp.68 (1996), 661-681. · Zbl 0859.11044  Cheney, E.W., Introduction to approximation theory, McGraw-Hill, New York, 1966. · Zbl 0161.25202  Chudnovsky, G., Number theoretic applications of polynomials with rational coefficients defined by extremality conditions, in Arithmetic and Geometry, M.Artin and J.Tate, Editors, vol. 1, Birkhaüser, Boston, 1983, 61-105. · Zbl 0547.10029  Davie, A.M. and Smyth, C.J., On a limiting fractal measure defined by conjugate algebraic integers, Publications Math. d’Orsay (1987-88), 93-103. · Zbl 0692.12002  Fekete, M., Über die Verteilung der Wurzelen bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math Zeit.17 (1923), 228-249. · JFM 49.0047.01  Ferguson, Le Baron O., Approximation by polynomials with integral coefficients, AMS, Rhode Island, 1980. · Zbl 0441.41003  Flammang, V., Sur la longueur des entiers algébriques totalement positifs, J. Number Th.54 (1995), 60-72. · Zbl 0831.11057  Flammang, V., Sur le diamètre transfini entier d’un intervalle à extrémités rationnelles, Ann. Inst. Fourier Grenoble45 (1995), 779-793. · Zbl 0826.41009  Flammang, V., Mesures de polynômes. Application au diamètre transfini entier, Thèse, Univ. de Metz, 1994. · Zbl 0842.26011  Golusin, G.M., Geometric theory of functions of a complex variable26 (1969), AMS Translations of Mathematical Monographs. · Zbl 0183.07502  Gorškov, D.S., On the distance from zero on the interval [0,1] of polynomials with integral coefficients (Russian), Proceedings of the Third All Union Mathematical congress (Moscow1956), vol. 4, Akad. Nauk. SSSR, 1959, 5-7.  Habsieger, L. and Salvy, B., On integer Chebyshev polynomials (1995), preprint A2X n° 95-21, Université Bordeaux I. · Zbl 0911.11033  Langevin, M., Diamètre transfini entier d’un intervalle à extrémités rationnelles (d’après F.Amoroso), preprint.  Montgomery, H.L., Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS84, Amer. Math. Soc., Providence, R.I., 1994. · Zbl 0814.11001  Rhin, G., Approximants de Padé et mesures effectives d’irrationalité, Séminaire de Théorie des nombres de Paris1985-86, C. Goldstein(Ed.), vol. 71, Progress in Math., Birkhäuser, 155-164. · Zbl 0632.10034  Rhin, G., Seminar, Pisa, 1989, unpublished.  Smyth, C.J., On the measure of totally real algebraic integers, J. Aust. Math. Soc. (Ser. A) 30 (1980), 137-149. · Zbl 0457.12001  Smyth, C.J., The mean value of totally real algebraic integers, Math. Comp.42 (1984), 663-681. · Zbl 0536.12006  Smyth, C.J., Totally positive algebraic integers of small trace, Ann. Inst. Fourier Grenoble34 (1984), 1-28. · Zbl 0534.12002  Steinmetz, N., Rational iteration: complex analytical dynamical systems, vol. 16, de Gruyter Studies in Mathematics, Berlin, 1993. · Zbl 0773.58010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.