Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Reciprocal polynomials having small measure

Author: David W. Boyd
Journal: Math. Comp. 35 (1980), 1361-1377
MSC: Primary 30C15; Secondary 12-04, 26C05, 65D20
MathSciNet review: 583514
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The measure of a monic polynomial is the product of the absolute value of the roots which lie outside and on the unit circle. We describe an algorithm, based on the root-squaring method of Graeffe, for finding all polynomials with integer coefficients whose measures and degrees are smaller than some previously given bounds. Using the algorithm, we find all such polynomials of degree at most 16 whose measures are at most 1.3. We also find all polynomials of height 1 and degree at most 26 whose measures satisfy this bound. Our results lend some support to Lehmer's conjecture. In particular, we find no noncyclotomic polynomial whose measure is less than the degree 10 example given by Lehmer in 1933.

References [Enhancements On Off] (What's this?)

  • [1] Erwin H. Bareiss, Resultant procedure and the mechanization of the Graeffe process, J. Assoc. Comput. Mach. 7 (1960), 346–386. MR 0119416
  • [2] David W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977), no. 2, 315–328. MR 0453692
  • [3] David W. Boyd, Variations on a theme of Kronecker, Canad. Math. Bull. 21 (1978), no. 2, 129–133. MR 0485771
  • [4] D. W. BOYD, "Pisot numbers and the width of meromorphic functions." (Privately circulated manuscript.)
  • [5] 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
  • [6] R. L. Duncan, Some inequalities for polynomials, Amer. Math. Monthly 73 (1966), 58–59. MR 0197690
  • [7] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, 10.2307/1968172
  • [8] Kurt Mahler, Lectures on transcendental numbers, Lecture Notes in Mathematics, Vol. 546, Springer-Verlag, Berlin-New York, 1976. MR 0491533
  • [9] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
  • [10] D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York-Berlin, 1970. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165. MR 0274686
  • [11] A. M. Ostrowski, On an inequality of J. Vicente Gonçalves, Univ. Lisboa Revista Fac. Ci. A (2) 8 (1960), 115–119. MR 0145049
  • [12] 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
  • [13] C. L. Stewart, On a theorem of Kronecker and a related question of Lehmer, Séminaire de Théorie des Nombres 1977–1978, CNRS, Talence, 1978, pp. Exp. No. 7, 11. MR 550267

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 30C15, 12-04, 26C05, 65D20

Retrieve articles in all journals with MSC: 30C15, 12-04, 26C05, 65D20

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society