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Mathematics of Computation
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
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1980-0583514-9
PII: S 0025-5718(1980)0583514-9
Article copyright: © Copyright 1980 American Mathematical Society