Reciprocal polynomials having small measure
Author:
David W. Boyd
Journal:
Math. Comp. 35 (1980), 13611377
MSC:
Primary 30C15; Secondary 1204, 26C05, 65D20
MathSciNet review:
583514
Fulltext PDF Free Access
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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 rootsquaring 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/S00255718198005835149
PII:
S 00255718(1980)05835149
Article copyright:
© Copyright 1980
American Mathematical Society
