Reciprocal polynomials having small measure. II
Reciprocal polynomials having small measure. II
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DOI：
10.1090/s0025-5718-1989-0968149-6
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发表时间：
1980-01
影响因子：
2
通讯作者：
D. Boyd
中科院分区：
文献类型：
--
作者：
D. Boyd
一元多项式的测度是位于单位圆之外和单位圆上的根的绝对值的乘积。我们描述了一种基于 Graeffe 的平方根方法的算法，用于查找所有具有整数系数且其度量和次数小于先前给定界限的多项式。使用该算法，我们找到所有此类次数最多为 16 且度量最多为 1.3 的多项式。我们还找到所有高度为 1、次数最多为 26 且其度量满足此界限的多项式。我们的结果为莱默的猜想提供了一些支持。特别是，我们没有发现任何非分圆多项式的测度小于 Lehmer 在 1933 年给出的 10 次示例。
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.