The maximal modulus of an algebraic integer
Author:
David W. Boyd
Journal:
Math. Comp. 45 (1985), 243249, S17
MSC:
Primary 11R06
MathSciNet review:
790657
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Abstract: The maximal modulus of an algebraic integer is the absolute value of its largest conjugate. We compute the minimum of the maximal modulus of all algebraic integers of degree d which are not roots of unity, for d at most 12. The computations suggest that the minimum is never attained for a reciprocal algebraic integer. The truth of this conjecture would show that the conjecture of Schinzel and Zassenhaus follows from a theorem of Smyth. We further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree d which are not roots of unity, for d at most 16. Our computations strongly suggest that the best constant in the conjecture of Schinzel and Zassenhaus is 1.5 , where is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers.
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 [1]
 D. W. Boyd, "Reciprocal polynomials having small measure," Math. Comp., v. 35, 1980, pp. 13611377. MR 583514 (82a:30005)
 [2]
 E. Dobrowolski, "On the maximal modulus of conjugates of an algebraic integer," Bull. Acad. Polon. Sci., v. 26, 1978, pp. 291292. MR 0491585 (58:10811)
 [3]
 E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial," Acta Arith., v. 34, 1979, pp. 391401. MR 543210 (80i:10040)
 [4]
 L. Kronecker, "Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten," J. Reine Angew. Math., v. 53, 1857, pp. 173175.
 [5]
 D. H. Lehmer, "Factorization of certain cyclotomic functions," Ann. of Math. (2), v. 34, 1933, pp. 461479. MR 1503118
 [6]
 D. Lind, "Entropies and factorizations of topological Markov shifts," Bull. Amer. Math. Soc. (N.S.), v. 9, 1983, pp. 219222. MR 707961 (84j:54010)
 [7]
 D. Lind, "The entropies of topological Markov shifts and a related class of algebraic integers," Ergodic Theory Dynamical Systems, v. 4, 1984, pp. 283300. MR 766106 (86c:58092)
 [8]
 W. Ljundggren, "On the irreducibility of certain trinomials and quadrinomials," Math. Scand., v. 8, 1960, pp. 6570. MR 0124313 (23:A1627)
 [9]
 A. Schinzel & H. Zassenhaus, "A refinement of two theorems of Kronecker," Michigan Math. J., v. 12, 1965, pp. 8184. MR 0175882 (31:158)
 [10]
 C. J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer," Bull London Math. Soc., v. 3, 1971, pp. 169175. MR 0289451 (44:6641)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507906578
PII:
S 00255718(1985)07906578
Keywords:
Algebraic integer,
maximal modulus,
SchinzelZassenhaus conjecture,
Perron numbers,
Smyth's theorem,
Newton's formulas
Article copyright:
© Copyright 1985
American Mathematical Society
