Using an idea of Doug Lind, we give a lower bound for the Perron–Frobenius degree of a Perron number that is not totally real, in terms of the layout of its Galois conjugates in the complex plane. As an application, we prove that there are cubic Perron numbers whose Perron–Frobenius degrees are arbitrary large, a result known to Lind, McMullen and Thurston. A similar result is proved for bi-Perron numbers.

中文翻译：

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Perron 数的 Perron–Frobenius 度的下界

使用 Doug Lind 的想法，根据复平面中伽罗瓦共轭的布局，我们给出了不完全真实的 Perron 数的 Perron-Frobenius 度的下界。作为应用，我们证明存在 Perron-Frobenius 度数任意大的三次 Perron 数，这是 Lind、McMullen 和 Thurston 已知的结果。对于 bi-Perron 数，也证明了类似的结果。