自反演多项式零点的单模性
Unimodularity of zeros of self-inversive polynomials
摘要：
推广了Cohn给出的自反演多项式的零点都在单位圆上的一个充要条件。我们的定理包含了Lakatos，Losonczi和Schinzel提出的一些充分条件。我们应用我们的结果来研究一个多项式家庭密切相关的Ramanujan多项式，最近推出的枪，Murty和Rath，并研究了Murty，史密斯和王和Lalín和罗杰斯。我们证明了这个族中的所有多项式在单位圆上都有零点，这是Lalín和Rogers在计算证据上证明的结果。
英文摘要：
We generalise a necessary and sufficient condition given by Cohn for all the zeros of a self-inversive polynomial to be on the unit circle. Our theorem implies some sufficient conditions found by Lakatos, Losonczi and Schinzel. We apply our result to the study of a polynomial family closely related to Ramanujan polynomials, recently introduced by Gun, Murty and Rath, and studied by Murty, Smyth and Wang and Lalín and Rogers. We prove that all polynomials in this family have their zeros on the unit circle, a result conjectured by Lalín and Rogers on computational evidence.
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