AI 中文总结
本文完成了多项式代数F[x]上任意权的单射Rota-Baxter算子的分类，明确了权为1时不同特征域下此类算子的具体形式。
AI 中文摘要
我们刻画了多项式代数F[x]上所有权为1的单射Rota-Baxter算子R。当域F的特征char F=p>0时，唯一的此类算子为R=-id；当char F=0时，此类算子要么为R=-id，要么在与F[x]的自同构共轭的意义下，满足R(1)=x，且该等式唯一确定R。结合已知的权为0的情形，这完成了F[x]上任意权的单射Rota-Baxter算子的分类。
英文摘要
We describe all injective Rota-Baxter operators $R$ of weight 1 on the polynomial algebra $F[x]$. When $\mathrm{char}\,F = p>0$, the only one is $R=-\mathrm{id}$. When $\mathrm{char}\,F = 0$, we have either $R = -\mathrm{id}$ or, up to conjugation with automorphisms of $F[x]$, $R(1) = x$, and $R$ is uniquely defined via this equality. Together with the known weight-zero case, this completes the classification of injective Rota-Baxter operators of any weight on $F[x]$.
Comments6 p.; Aristotle was used
