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} ; J&ouml ; 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 mmlsi1" class="mathmlsrc">mulatext 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}mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi>Pmi><mo>,mo><mi>Qmi><mo>∈mo><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub><mo stretchy="false">[mo><mi>Xmi><mo stretchy="false">]mo><mo>∖mo><mo stretchy="false">{mo><mn>0mn><mo stretchy="false">}mo>math> be two coprime polynomials over the finite field mmlsi124" class="mathmlsrc">mulatext 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_qmathContainer hidden">mathCode"><math altimg="si124.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub>math> with mmlsi11" class="mathmlsrc">mulatext 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⁡QmathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><mi mathvariant="normal">degmi><mo>⁡mo><mi>Pmi><mo>>mo><mi mathvariant="normal">degmi><mo>⁡mo><mi>Qmi>math>. We represent each polynomial m>w  m> over mmlsi124" class="mathmlsrc">mulatext 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_qmathContainer hidden">mathCode"><math altimg="si124.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub>math> by using a rational m>base  m>mmlsi5" class="mathmlsrc">mulatext 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/QmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi>Pmi><mo stretchy="false">/mo><mi>Qmi>math> and m>digits  m>mmlsi6" class="mathmlsrc">mulatext 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]mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><msub><mrow><mi>smi>mrow><mrow><mi>imi>mrow>msub><mo>∈mo><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub><mo stretchy="false">[mo><mi>Xmi><mo stretchy="false">]mo>math> satisfying mmlsi644" class="mathmlsrc">mulatext 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⁡PmathContainer hidden">mathCode"><math altimg="si644.gif" overflow="scroll"><mi mathvariant="normal">degmi><mo>⁡mo><msub><mrow><mi>smi>mrow><mrow><mi>imi>mrow>msub><mo><mo><mi mathvariant="normal">degmi><mo>⁡mo><mi>Pmi>math>. m>Digit expansions  m> of this type are also defined for formal Laurent series over mmlsi124" class="mathmlsrc">mulatext 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_qmathContainer hidden">mathCode"><math altimg="si124.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub>math>. We prove uniqueness and automatic properties of these expansions. Although the m>ω  m>-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 mmlsi193" class="mathmlsrc">mulatext 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]mathContainer hidden">mathCode"><math altimg="si193.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Fmi>mrow><mrow><mi>qmi>mrow>msub><mo stretchy="false">[mo><mi>Xmi><mo stretchy="false">]mo>math>. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's 3/2-problem.