The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of Möbius transformations – often called Cayley transformations – that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail several classes of polynomials with symmetric zeros as, for instance, the cases of self-conjugate, self-adjoint, self-inversive, self-reciprocal and skew-reciprocal polynomials. Finally, an application of this method to Salem polynomials and to polynomials with small Mahler measure is also discussed.

Sturm很久以前就解决了计算实多项式的实零数的经典问题。但是，对多项式在单位圆上具有的零进行计数的类似问题仍然是一个未解决的问题。在本文中，我们表明，可以通过使用一对合适的Möbius变换（通常称为Cayley变换）将第二个问题简化为第一个问题，该变换具有将单位圆映射到实线上，反之亦然的特性。尽管该方法适用于任意复数多项式，但我们详细讨论了几类具有对称零的多项式，例如，自共轭，自伴随，自反，自反和偏向倒数多项式的情况。最后，