On global -forms

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• 作者：Xiang-dong Hou ^{xhou@usf.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11T06 ; 11R27 ; 14E07
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：307-325
• 全文大小：399 K

文摘

Let F_q${\mathbb{F}}_q$ be a finite field with $\text{char}{\mathbb{F}}_q=p$ and n>0$n>0$ an integer with gcd(n,log_p⁡q)=1$\text{gcd}(n,{\log }_{p}q)=1$. Let ${()}^{⁎}:{\mathbb{F}}_q({\mathtt{x}}_0,\dots ,{\mathtt{x}}_{n-1})\to {\mathbb{F}}_q({\mathtt{x}}_0,\dots ,{\mathtt{x}}_{n-1})$ be the F_q${\mathbb{F}}_q$-monomorphism defined by ${\mathtt{x}}_i^{⁎}={\mathtt{x}}_{i+1}$ for 0≤i<n−1$0\le i<n-1$ and ${\mathtt{x}}_{n-1}^{⁎}={\mathtt{x}}_0^q$. For 05f4d4bf8ad73d2a9fa07e38d9b1aa60" title="Click to view the MathML source">f,g∈F_q(x₀,…,x_n−1)∖F_q$f,g\in {\mathbb{F}}_q({\mathtt{x}}_0,\dots ,{\mathtt{x}}_{n-1})\setminus {\mathbb{F}}_q$, define f∘g=f(g,g^⁎,…,g^(n−1)⁎)$f\circ g=f(g,g^{⁎},\dots ,g^{(n-1)⁎})$. Then $({\mathbb{F}}_q({\mathtt{x}}_0,\dots ,{\mathtt{x}}_{n-1})\setminus {\mathbb{F}}_q,\circ )$ is a monoid whose invertible elements are called global P$\mathcal{P}$-forms. Global P$\mathcal{P}$-forms were first introduced by H. Dobbertin in 2001 with q=2$q=2$ to study a certain type of permutation polynomials of F_2^m${\mathbb{F}}_{2^m}$ with 0594106a757be70d5c75339baddbc5d4" title="Click to view the MathML source">gcd(m,n)=1$\text{gcd}(m,n)=1$; global P$\mathcal{P}$-forms with 0575" title="Click to view the MathML source">q=p$q=p$ for an arbitrary prime p   were considered by W. More in 2005. In this paper, we discuss some fundamental questions about global P$\mathcal{P}$-forms, some of which are answered and others remain open.