Let \(P(x)=a_0x^d+a_1x^{d-1}+\cdots +a_d \in {{\mathbb {Q}}}[x]\) be a polynomial of degree \(d \ge 2\), and let \(x_n\), \(n=0,1,2,\ldots \), be a sequence of integers satisfying \(x_{n+1}=P(x_n)\) for \(n \ge 0\) and \(x_n \rightarrow \infty \) as \(n \rightarrow \infty \). Then, by a recent result of Wagner and Ziegler, \(\alpha =\lim _{n\rightarrow \infty } x_n^{d^{-n}}>1\) is either an integer or an irrational number, and \(x_n\) is approximately \(a_0^{-1/(d-1)} \alpha ^{d^n}-a_1/(da_0)\). Under assumption \(a_0^{1/(d-1)} \in {{\mathbb {Q}}}\) on the leading coefficient \(a_0\) of P, we completely characterize all the cases when the limit \(\alpha \) is an algebraic number. Our results imply that \(\alpha \) can be an integer, a quadratic Pisot unit with \(\alpha ^{-1}\) being its conjugate over \({{\mathbb {Q}}}\), or a transcendental number. In most cases \(\alpha \) is transcendental. For each \(d \ge 2\) all the polynomials P of degree d for which \(\alpha \) is an integer or a quadratic Pisot unit are described explicitly. The main theorem implies that several constants related to sequences that appear in a paper of Aho and Sloane and in the online Encyclopedia of Integer Sequences are transcendental.

设\(P(x)=a_0x^d+a_1x^{d-1}+\cdots +a_d \in {{\mathbb {Q}}}[x]\)是一个多项式\(d \ge 2\)，让\(x_n\) , \(n=0,1,2,\ldots \)是满足\(x_{n+1}=P(x_n)\)对于\ (n \ge 0\)和\(x_n \rightarrow \infty \)为\(n \rightarrow \infty \)。然后，根据瓦格纳和齐格勒的最新结果，\(\alpha =\lim _{n\rightarrow \infty } x_n^{d^{-n}}>1\)是整数或无理数，并且\(x_n\)大约是\(a_0^{-1/(d-1)} \alpha ^{d^n}-a_1/(da_0)\)。假设下\(a_0^{1/(d-1)} \in {{\mathbb {Q}}}\)在P的前导系数\(a_0\)上，我们完全刻画了极限\(\ alpha \)是一个代数数。我们的结果意味着\(\alpha \)可以是一个整数，一个二次 Pisot 单位，其中\(\alpha ^{-1}\)是它在\({{\mathbb {Q}}}\)上的共轭，或者一个超越数。在大多数情况下\(\alpha \)是超越的。对于每个\(d \ge 2\)所有的多项式P的d 次，其中\(\alpha \)是一个整数或一个二次 Pisot 单位被明确描述。主要定理意味着与出现在 Aho 和 Sloane 的论文以及在线整数序列百科全书中的序列相关的几个常数是超越的。