Inverse stability of linearized polynomials over ffniteffelds
来源：
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发布时间：2026-04-20
报告时间：
2026年4月17日（周五）下午14:00-15:00
报告地点：
国交2号楼315会议室
报告人：
程开敏
报告人单位：
西华师范大学
邀请人：
刘宏伟
摘要：
For a polynomial $\phi\in\mathbb{F}_q[X]$, let $d_{n,\phi}(X)$ denote the denominator of the $n$th iterate of $1/\phi(X)$. We call $\phi$ \emph{inversely stable} if the polynomials $d_{n,\phi}$ are pairwise distinct and irreducible over $\mathbb{F}_q$ for all $n\ge 1$. In this talk, we shall classify the monic linearized polynomials $L(X)+b$ with this property. We prove a rigidity theorem showing that if $L$ is separable linearized of degree $p^r$ and $L(X)+b$ is irreducible, then after adjoining one root no further translate with the same linearized part can remain irreducible unless $r=1$. Hence inverse stability in the monic linearized family occurs only in degree $p$. For $X^p+aX+b$ we obtain an explicit criterion in terms of absolute traces of a recursively defined sequence, extending the Artin--Schreier case to the For a polynomial $\phi\in\mathbb{F}_q[X]$, let $d_{n,\phi}(X)$ denote the denominator of the $n$th iterate of $1/\phi(X)$. We call $\phi$ \emph{inversely stable} if the polynomials $d_{n,\phi}$ are pairwise distinct and irreducible over $\mathbb{F}_q$ for all $n\ge 1$. In this talk, we shall classify the monic linearized polynomials $L(X)+b$ with this property. We prove a rigidity theorem showing that if $L$ is separable linearized of degree $p^r$ and $L(X)+b$ is irreducible, then after adjoining one root no further translate with the same linearized part can remain irreducible unless $r=1$. Hence inverse stability in the monic linearized family occurs only in degree $p$. For $X^p+aX+b$ we obtain an explicit criterion in terms of absolute traces of a recursively defined sequence, extending the Artin--Schreier case to the full degree-$p$ family. The recurrence yields a finite dynamical system and therefore a finite stopping rule for deciding inverse stability...full degree-$p$ family. The recurrence yields a finite dynamical system and therefore a finite stopping rule for deciding inverse stability...
来源：
发布时间：2026-04-20
报告时间：
