Abhishek Bharadwaj, Aprameyo Pal, Veekesh Kumar, R. Thangadurai
{"title":"Sufficient conditions for a problem of Polya","authors":"Abhishek Bharadwaj, Aprameyo Pal, Veekesh Kumar, R. Thangadurai","doi":"10.1090/proc/16826","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a non-zero algebraic number. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the Galois closure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q left-parenthesis alpha right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}(\\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with Galois group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\bar {\\mathbb {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the algebraic closure of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this article, among the other results, we prove the following. <italic>If <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f element-of ModifyingAbove double-struck upper Q With bar left-bracket upper G right-bracket\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">¯</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f\\in \\bar {\\mathbb {Q}}[G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-zero element of the group ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove double-struck upper Q With bar left-bracket upper G right-bracket\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">¯</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\bar {\\mathbb {Q}}[G]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a given algebraic number such that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis alpha Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f(\\alpha ^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a non-zero algebraic integer for infinitely many natural numbers <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha\"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding=\"application/x-tex\">\\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an algebraic integer.</italic> This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni <italic>et al.</italic> [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16826","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let α\alpha be a non-zero algebraic number. Let KK be the Galois closure of Q(α)\mathbb {Q}(\alpha ) with Galois group GG and Q¯\bar {\mathbb {Q}} be the algebraic closure of Q\mathbb {Q}. In this article, among the other results, we prove the following. If f∈Q¯[G]f\in \bar {\mathbb {Q}}[G] is a non-zero element of the group ring Q¯[G]\bar {\mathbb {Q}}[G] and α\alpha is a given algebraic number such that f(αn)f(\alpha ^n) is a non-zero algebraic integer for infinitely many natural numbers nn, then α\alpha is an algebraic integer. This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem.