The inverse stability of Artin-Schreier polynomials over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912390976634880 |
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| author | Cheng, Kaimin |
| author_facet | Cheng, Kaimin |
| contents | Let $p$ be a prime number and $q$ a power of $p$. Let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $φ(X)\in\mathbb{F}_q[X]$, let $d_{n,φ}(X)$ denote the denominator of the $n$th iterate of $\frac{1}{φ(X)}$. The polynomial $φ(X)$ is said to be inversely stable over $\mathbb{F}_q$ if all polynomials $d_{n,φ}(X)$ are irreducible polynomial over $\mathbb{F}_q$ and distinct. In this paper, we characterize a class of inversely stable polynomials over $\mathbb{F}_q$. More precisely, for $φ(X)=X^{p^t}+aX+b\in\mathbb{F}_q[X]$ with $t$ being a positive integer, we provide a sufficient and necessary condition for $φ(X)$ to be inversely stable over $\mathbb{F}_q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The inverse stability of Artin-Schreier polynomials over finite fields Cheng, Kaimin Number Theory 11T06 Let $p$ be a prime number and $q$ a power of $p$. Let $\mathbb{F}_q$ be the finite field with $q$ elements. For a positive integer $n$ and a polynomial $φ(X)\in\mathbb{F}_q[X]$, let $d_{n,φ}(X)$ denote the denominator of the $n$th iterate of $\frac{1}{φ(X)}$. The polynomial $φ(X)$ is said to be inversely stable over $\mathbb{F}_q$ if all polynomials $d_{n,φ}(X)$ are irreducible polynomial over $\mathbb{F}_q$ and distinct. In this paper, we characterize a class of inversely stable polynomials over $\mathbb{F}_q$. More precisely, for $φ(X)=X^{p^t}+aX+b\in\mathbb{F}_q[X]$ with $t$ being a positive integer, we provide a sufficient and necessary condition for $φ(X)$ to be inversely stable over $\mathbb{F}_q$. |
| title | The inverse stability of Artin-Schreier polynomials over finite fields |
| topic | Number Theory 11T06 |
| url | https://arxiv.org/abs/2412.04985 |