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arXiv · 2304.03992

On stable polynomials of degrees $2,3,4$

Tong Lin·Qiang Wang
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Abstract

Let $q$ be a prime power. We construct stable polynomials of the form $b^{m-1}(x+a)^m+c(x+a)+d$ over a finite field $\mathbb{F}_{q}$ for $m=2,3,4$ by Capelli's lemma. When $m=3$ and $q$ is even, we confirm the conjecture of Ahmadi and Monsef-Shokri [2] that the polynomial $f(x) = x^3 + x^2 + 1$ is stable over $\mathbb{F}_{2}$. Moreover, when $m=2$ and $q\equiv 1\pmod{4}$, we improve a lower bound of the number of quadratic stable polynomials by Gom\'ez-P\'erez and Nicol\'as [4].

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Tong Lin, Qiang Wang. 2023-04-08. On stable polynomials of degrees $2,3,4$. https://arxiv.org/abs/2304.03992

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