The mean values of logarithms of algebraic integers

Artūras Dubickas

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 2, page 301-313
  • ISSN: 1246-7405

Abstract

top
Let α be an algebraic integer of degree d with conjugates α 1 = α , α 2 , ⋯ , α d . In the paper we give a lower bound for the mean value M p ( α ) = 1 d ∑ i = 1 d | log | α i | | p p when α is not a root of unity and p > 1 .

How to cite

top

Dubickas, Artūras. "The mean values of logarithms of algebraic integers." Journal de théorie des nombres de Bordeaux 10.2 (1998): 301-313. <http://eudml.org/doc/248169>.

@article{Dubickas1998,
abstract = {Let $\alpha $ be an algebraic integer of degree $d$ with conjugates $\alpha _1 = \alpha , \alpha _2, \dots , \alpha _d$. In the paper we give a lower bound for the mean value\begin\{equation*\} M\_p(\alpha )=\@root p \of \{\frac\{1\}\{d\} \sum ^\{d\}\_\{i=1\}|\log |\alpha \_i||^p\}\end\{equation*\}when $\alpha $ is not a root of unity and $p &gt; 1$.},
author = {Dubickas, Artūras},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Mahler's measure; logarithms of algebraic integers; mean values of logarithms},
language = {eng},
number = {2},
pages = {301-313},
publisher = {Université Bordeaux I},
title = {The mean values of logarithms of algebraic integers},
url = {http://eudml.org/doc/248169},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Dubickas, Artūras
TI - The mean values of logarithms of algebraic integers
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 2
SP - 301
EP - 313
AB - Let $\alpha $ be an algebraic integer of degree $d$ with conjugates $\alpha _1 = \alpha , \alpha _2, \dots , \alpha _d$. In the paper we give a lower bound for the mean value\begin{equation*} M_p(\alpha )=\@root p \of {\frac{1}{d} \sum ^{d}_{i=1}|\log |\alpha _i||^p}\end{equation*}when $\alpha $ is not a root of unity and $p &gt; 1$.
LA - eng
KW - Mahler's measure; logarithms of algebraic integers; mean values of logarithms
UR - http://eudml.org/doc/248169
ER -

References

top
  1. [1] D. Bertrand, Duality on tori and multiplicative dependence relations. J. Austral. Math. Soc. (to appear). Zbl0886.11035MR1433209
  2. [2] P.E. Blanksby, H.L. Montgomery, Algebraic integers near the unit circle. Acta Arith.18 (1971), 355-369. Zbl0221.12003MR296021
  3. [3] D.C. Cantor, E.G. Straus, On a conjecture of D.H.Lehmer. Acta Arith.42 (1982), 97-100. Zbl0504.12002MR679001
  4. [4] E. Dobrowolski, On a question of Lehmer and the number of irreducibile factors of a polynomial. Acta Arith.34 (1979), 391-401. Zbl0416.12001MR543210
  5. [5] A. Dubickas, On a conjecture of Schinzel and Zassenhaus. Acta Arith.63 (1993), 15-20. Zbl0777.11039MR1201616
  6. [6] A. Dubickas, On the average difference between two conjugates of an algebraic number. Liet. Matem. Rink.35 (1995), 415-420. Zbl0862.11058MR1407440
  7. [7] M. Langevin, Solution des problèmes de Favard. Ann. Inst. Fourier38 (1988), no. 2, 1-10. Zbl0634.12002MR949008
  8. [8] D.H. Lehmer, Factorization of certain cyclotomic functions. Ann. of Math.34 (1933), 461-479. Zbl0007.19904MR1503118JFM59.0933.03
  9. [9] R. Louboutin, Sur la mesure de Mahler d'un nombre algébrique. C.R.Acad. Sci. Paris296 (1983), 707-708. Zbl0557.12001MR706663
  10. [10] E.M. Matveev, A connection between Mahler measure and the discriminant of algebraic numbers. Matem. Zametki59 (1996), 415-420 (in Russian). Zbl0879.11056MR1399967
  11. [11] M. Meyer, Le problème de Lehmer: méthode de Dobrowolski et lemme de Siegel "à la Bombieri-Vaaler". Publ. Math. Univ. P. et M. Curie (Paris VI), 90, Problèmes Diophantiens (1988-89), No.5, 15 p. 
  12. [12] M. Mignotte, M. Waldschmidt, On algebraic numbers of small height: linear forms in one logarithm. J. Number Theory47 (1994), 43-62. Zbl0801.11033MR1273455
  13. [13] A. Schinzel, H. Zassenhaus, A refinement of two theorems of Kronecker. Michigan Math. J.12 (1965), 81-85. Zbl0128.03402MR175882
  14. [14] I. Schur, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Zeitschrift1 (1918), 377-402. Zbl46.0128.03MR1544303JFM46.0128.03
  15. [15] C.L. Siegel, The trace of totally positive and real algebraic integers. Ann. of Math.46 (1945), 302-312. Zbl0063.07009MR12092
  16. [16] C.J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer. Bull. London Math. Soc.3 (1971), 169-175. Zbl0235.12003MR289451
  17. [17] C.J. Smyth, The mean values of totally real algebraic integers. Math. Comp.42 (1984), 663-681. Zbl0536.12006MR736460
  18. [18] C.L. Stewart, Algebraic integers whose conjugates lie near the unit circle. Bull. Soc. Math. France106 (1978), 169-176. Zbl0396.12002MR507748
  19. [19] P. Voutier, An effective lower bound for the height of algebraic numbers. Acta Arith.74 (1996), 81-95. Zbl0838.11065MR1367580

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.