The maximal modulus of an algebraic integer

Author:
David W. Boyd

Journal:
Math. Comp. **45** (1985), 243-249, S17

MSC:
Primary 11R06

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790657-8

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.

**[1]**David W. Boyd,*Reciprocal polynomials having small measure*, Math. Comp.**35**(1980), no. 152, 1361–1377. MR**583514**, https://doi.org/10.1090/S0025-5718-1980-0583514-9**[2]**Edward Dobrowolski,*On the maximal modulus of conjugates of an algebraic integer*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**26**(1978), no. 4, 291–292 (English, with Russian summary). MR**0491585****[3]**E. Dobrowolski,*On a question of Lehmer and the number of irreducible factors of a polynomial*, Acta Arith.**34**(1979), no. 4, 391–401. MR**543210**, https://doi.org/10.4064/aa-34-4-391-401**[4]**L. Kronecker, "Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten,"*J. Reine Angew. Math.*, v. 53, 1857, pp. 173-175.**[5]**D. H. Lehmer,*Factorization of certain cyclotomic functions*, Ann. of Math. (2)**34**(1933), no. 3, 461–479. MR**1503118**, https://doi.org/10.2307/1968172**[6]**D. A. Lind,*Entropies and factorizations of topological Markov shifts*, Bull. Amer. Math. Soc. (N.S.)**9**(1983), no. 2, 219–222. MR**707961**, https://doi.org/10.1090/S0273-0979-1983-15162-5**[7]**D. A. Lind,*The entropies of topological Markov shifts and a related class of algebraic integers*, Ergodic Theory Dynam. Systems**4**(1984), no. 2, 283–300. MR**766106**, https://doi.org/10.1017/S0143385700002443**[8]**Wilhelm Ljunggren,*On the irreducibility of certain trinomials and quadrinomials*, Math. Scand.**8**(1960), 65–70. MR**0124313**, https://doi.org/10.7146/math.scand.a-10593**[9]**A. Schinzel and H. Zassenhaus,*A refinement of two theorems of Kronecker*, Michigan Math. J.**12**(1965), 81–85. MR**0175882****[10]**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. MR**0289451**, https://doi.org/10.1112/blms/3.2.169

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790657-8

Keywords:
Algebraic integer,
maximal modulus,
Schinzel-Zassenhaus conjecture,
Perron numbers,
Smyth's theorem,
Newton's formulas

Article copyright:
© Copyright 1985
American Mathematical Society