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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=8f0bc7ea82a970bfd09c55d99f04e746" title="Click to view the MathML source">P,Q∈F_q[X]∖{0}class="mathContainer hidden">class="mathCode">$P,Q\in {\mathbb{F}}_q[X]\setminus \{0\}$ be two coprime polynomials over the finite field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si11.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=4d8dafdf022fb32645b67ac6b75697f2" title="Click to view the MathML source">deg⁡P>deg⁡Qclass="mathContainer hidden">class="mathCode">$\mathrm{deg}P>\mathrm{deg}Q$. We represent each polynomial w   over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$ by using a rational base  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=e18775548425ff80b18ef905fe9327bb" title="Click to view the MathML source">P/Qclass="mathContainer hidden">class="mathCode">$P/Q$ and digits  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=3c0274cac2d9c567c3d57a4aed0d93ac" title="Click to view the MathML source">s_i∈F_q[X]class="mathContainer hidden">class="mathCode">$s_i\in {\mathbb{F}}_q[X]$ satisfying class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si644.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=f8313c89bc82ca7a6cb85f0c012a3c6e" title="Click to view the MathML source">deg⁡s_i<deg⁡Pclass="mathContainer hidden">class="mathCode">$\mathrm{deg}{s}_i<\mathrm{deg}P$. Digit expansions   of this type are also defined for formal Laurent series over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si124.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=b9264fbb0cd6ddc57c9e30dc465fe92f" title="Click to view the MathML source">F_qclass="mathContainer hidden">class="mathCode">${\mathbb{F}}_q$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302049&_mathId=si193.gif&_user=111111111&_pii=S0022314X16302049&_rdoc=1&_issn=0022314X&md5=ab21ee811c9f8af161d6673fc9a73037" title="Click to view the MathML source">F_q[X]class="mathContainer hidden">class="mathCode">${\mathbb{F}}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's 3/2-problem.