## Polynomials with small Mahler measure

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- by Michael J. Mossinghoff PDF
- Math. Comp.
**67**(1998), 1697-1705 Request permission

## Abstract:

We describe several searches for polynomials with integer coefficients and small Mahler measure. We describe the algorithm used to test Mahler measures. We determine all polynomials with degree at most 24 and Mahler measure less than $1.3$, test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40, and check certain sparse polynomials with height 1 and degree as large as 181. We find a new limit point of Mahler measures near $1.309$, four new Salem numbers less than $1.3$, and many new polynomials with small Mahler measure. None has measure smaller than that of Lehmer’s degree 10 polynomial.## References

- Enrico Bombieri and Jeffrey D. Vaaler,
*Polynomials with low height and prescribed vanishing*, Analytic number theory and Diophantine problems (Stillwater, OK, 1984) Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 53–73. MR**1018369** - J. Hiller and D. K. Goodman,
*Realisation from partial sequences*, Electron. Lett.**7**(1971), 188–189. MR**452878**, DOI 10.1049/el:19710125 - David W. Boyd,
*Pisot and Salem numbers in intervals of the real line*, Math. Comp.**32**(1978), no. 144, 1244–1260. MR**491587**, DOI 10.1090/S0025-5718-1978-0491587-8 - David W. Boyd,
*Reciprocal polynomials having small measure*, Math. Comp.**35**(1980), no. 152, 1361–1377. MR**583514**, DOI 10.1090/S0025-5718-1980-0583514-9 - David W. Boyd,
*Speculations concerning the range of Mahler’s measure*, Canad. Math. Bull.**24**(1981), no. 4, 453–469. MR**644535**, DOI 10.4153/CMB-1981-069-5 - David W. Boyd,
*Reciprocal polynomials having small measure. II*, Math. Comp.**53**(1989), no. 187, 355–357, S1–S5. MR**968149**, DOI 10.1090/S0025-5718-1989-0968149-6 - 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**, DOI 10.4064/aa-34-4-391-401 - L. Kantorovitch,
*The method of successive approximations for functional equations*, Acta Math.**71**(1939), 63–97. MR**95**, DOI 10.1007/BF02547750 - D. H. Lehmer,
*Factorization of certain cyclotomic functions*, Ann. of Math. (2)**34**(1933), 461–479. - M. J. Mossinghoff,
*Algorithms for the determination of polynomials with small Mahler measure*, Ph.D. Thesis, The University of Texas at Austin, 1995. - M. J. Mossinghoff, C. G. Pinner, and J. D. Vaaler,
*Perturbing polynomials with all their roots on the unit circle*, Math. Comp. 67 (1998), 1707–1726. - Walter Gautschi (ed.),
*Mathematics of Computation 1943–1993: a half-century of computational mathematics*, Proceedings of Symposia in Applied Mathematics, vol. 48, American Mathematical Society, Providence, RI, 1994. Papers from the Symposium on Numerical Analysis and the Minisymposium on Computational Number Theory held in Vancouver, British Columbia, August 9–13, 1993. MR**1314838**, DOI 10.1090/psapm/048 - 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**289451**, DOI 10.1112/blms/3.2.169 - J. Stoer and R. Bulirsch,
*Introduction to numerical analysis*, Springer-Verlag, New York-Heidelberg, 1980. Translated from the German by R. Bartels, W. Gautschi and C. Witzgall. MR**557543**, DOI 10.1007/978-1-4757-5592-3

## Additional Information

**Michael J. Mossinghoff**- Affiliation: Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: mjm@math.appstate.edu
- Received by editor(s): May 20, 1997
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1697-1705 - MSC (1991): Primary 12--04; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-98-01006-0
- MathSciNet review: 1604391