Rational digit systems over finite fields and Christol's Theorem

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• 作者：Manuel Joseph C. Loquias^a ; ^b ; ^{mjcloquias@math.upd.edu.ph" class="auth_mail" title="E-mail the corresponding author} ; Mohamed Mkaouar^c ; ^{mohamed.mkaouar@fss.rnu.tn" class="auth_mail" title="E-mail the corresponding author} ; Klaus Scheicher^d ; ^{klaus.scheicher@boku.ac.at" class="auth_mail" title="E-mail the corresponding author} ; Jö ; rg M. Thuswaldner^a ; ^{joerg.thuswaldner@unileoben.ac.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11A63 ; 68Q70 ; secondary ; 11B85
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：358-390
• 全文大小：621 K

文摘

Let P,Q∈F_q[X]∖{0}$P,Q\in {\mathbb{F}}_q[X]\setminus \{0\}$ be two coprime polynomials over the finite field e30dc465fe92f" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ with deg⁡P>deg⁡Q$\mathrm{deg}P>\mathrm{deg}Q$. We represent each polynomial w   over e30dc465fe92f" title="Click to view the MathML source">F_q${\mathbb{F}}_q$ by using a rational base  18ef905fe9327bb" title="Click to view the MathML source">P/Q$P/Q$ and digits  s_i∈F_q[X]$s_i\in {\mathbb{F}}_q[X]$ satisfying deg⁡s_i<deg⁡P$\mathrm{deg}{s}_i<\mathrm{deg}P$. Digit expansions   of this type are also defined for formal Laurent series over e30dc465fe92f" title="Click to view the MathML source">F_q${\mathbb{F}}_q$. We prove uniqueness and automatic properties of these expansions. Although the ω  -language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over F_q[X]${\mathbb{F}}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's 3/2-problem.