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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
Our proof uses an absolutely irreducible curve over $\mathbb{F}_{q^n}$ and the number of rational points on it.
📌 3 Key Takeaways
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Objective & Hypothesis:Öz We consider the polynomials of the form $P(x)=x^k-\gamma \mathrm{Tr}(x)$ over $\mathbb{F}_{q^n}$ for $n\geq 2$.
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Key Findings & Data:We show that $P(x)$ is not a permutation of $\mathbb{F}_{q^n}$ in the case $\gcd(k, q^n-1)>1$.
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Conclusion & Impact:Our proof uses an absolutely irreducible curve over $\mathbb{F}_{q^n}$ and the number of rational points on it.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Anahtar Kelimeler Function fields · permutation polynomials · rational places
Özet
Öz We consider the polynomials of the form $P(x)=x^k-\gamma \mathrm{Tr}(x)$ over $\mathbb{F}_{q^n}$ for $n\geq 2$. We show that $P(x)$ is not a permutation of $\mathbb{F}_{q^n}$ in the case $\gcd(k, q^n-1)>1$. Our proof uses an absolutely irreducible curve over $\mathbb{F}_{q^n}$ and the number of rational points on it.
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Cite This Article
MEIDL, N. A. (2019). Curves over Finite Fields and Permutations of the Form $x^k-\gamma \mathrm{Tr}(x)$. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 1, 533–538. https://ajindex.com/articles/218069