关于Stern多项式算术性质的注记
A note on the arithmetic properties of Stern Polynomials
摘要：
我们研究了由$B_0 ( t ) =0,B_1 ( t ) =1$定义的Stern多项式，并对$n \geq 2$研究了递推关系$B_{2n}( t) =tB_{n}( t) ,$ $B_{2n+1}( t) =B_n( t) +B_{n+1}( t) $。证明了该多项式的所有可能的有理根都是$0,-1,-1/2,-1/3$。我们给出了$n$的完整表征，使得$deg( B_n) = deg( B_{n+1}) $和$deg( B_n) =deg( B_{n+1}) =deg( B_{n+2}) $。此外，我们还给出了关于倒易Stern多项式的一些结果。
英文摘要：
We investigate the Stern polynomials defined by $B_0 ( t ) =0,B_1 ( t ) =1$, and for $n \geq 2$ by the recurrence relations $B_{2n}( t) =tB_{n}( t) ,$ $B_{2n+1}( t) =B_n( t) +B_{n+1}( t) $. We prove that all possible rational roots of that polynomials are $0,-1,-1/2,-1/3$. We give complete characterization of $n$ such that $deg( B_n) = deg( B_{n+1}) $ and $deg( B_n) =deg( B_{n+1}) =deg( B_{n+2}) $. Moreover, we present some result concerning reciprocal Stern polynomials.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
