Tanaka, Takaaki

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Professor

Related Websites

External Links
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Career 【 Display / hide 】

  • 1996.04
    -
    1998.03

    日本学術振興会 ,特別研究員

  • 1998.04
    -
    2007.03

    慶應義塾大学（理工学部）,助手

  • 2001.02
    -
    2002.03

    フンボルト財団（ドイツ連邦共和国，ケルン大学） ,奨学研究員

  • 2006.04
    -
    2009.03

    慶應義塾大学（経済学部）,非常勤講師

  • 2007.04
    -
    2010.03

    慶應義塾大学（理工学部）,助教

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Academic Background 【 Display / hide 】

  • 1993.03

    Keio University, Faculty of Science and Engineering, 数理科学科

    University, Graduated

  • 1995.03

    Keio University, Graduate School, Division of Science and Engineeri, 数理科学専攻

    Graduate School, Completed, Master's course

  • 1998.03

    Keio University, Graduate School, Division of Science and Engineeri, 数理科学専攻

    Graduate School, Completed, Doctoral course

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Academic Degrees 【 Display / hide 】

  • 博士（理学）, Keio University, Coursework, 1998.03

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Licenses and Qualifications 【 Display / hide 】

  • 気象予報士, 1999.10

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Research Areas 【 Display / hide 】

  • Natural Science / Algebra (Analytic Number Theory)

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Research Keywords 【 Display / hide 】

  • algebraic independence

  • transcendental numbers

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Papers 【 Display / hide 】

  • Algebraic independence of the partial derivatives of certain functions with arbitrary number of variables

    Ide Haruki, Tanaka Taka-aki

    Indagationes Mathematicae (Indagationes Mathematicae)  34 ( 6 ) 1397 - 1418 2023.11

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  00193577

    　View Summary

    We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero components, is algebraically independent. In Section 2 of this paper, we develop a technique involving linear isomorphisms and infinite products to replace the algebraic independence of the values of functions in question with that of functions easier to deal with. In Sections 2 and 3, using the technique together with Mahler's method, we can reduce the algebraic independence of the infinite set mentioned above to the linear independence of certain rational functions modulo the rational function field of many variables. The latter one is solved by the discussions involving a certain valuation and a generic point in Sections 3 and 4.

  • Algebraic independence of the values of power series and their derivatives generated by linear recurrences

    Haruki Ide, Taka-aki Tanaka, Kento Toyama

    Tokyo J. Math (Tokyo Journal of Mathematics)  45 ( 2 ) 519 - 545 2022.12

    Research paper (scientific journal), Last author, Accepted,  ISSN  03873870

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    Using a descent method, we construct certain power series generated by linear recurrences, each of which possesses the following property: The infinite set consisting of all its values and all the values of its derivatives of any order, at any nonzero algebraic numbers within its domain of existence, is algebraically independent. The main theorems of this paper assert that
    the power series generated by a linear recurrence simpler than ever before, have this property.

  • On power series generated by simpler sequences and having strong algebraic independence properties

    TANAKA TAKA-AKI, KENTO TOYAMA

    RIMS Kôkyûroku (RIMS)  2222   219 - 236 2022.06

    Research paper (international conference proceedings), Joint Work, Lead author

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    0と1を係数とする冪級数にいて，1を係数とする項の指数から成る非負整数列を考える．そのような数列が等差数列と等比数列の和で表される場合について，当該冪級数の収束円内の0を除く任意の代数的数における値および任意の階数の微分係数をすべて併せた無限集合が代数的独立となる必要十分条件を記述した．

  • Algebraic independence of the values of a certain map defined on the set of orbits of the action of Klein four-group

    TANAKA TAKA-AKI

    RIMS Kôkyûroku (RIMS)  2131   177 - 187 2019.10

    Research paper (international conference proceedings), Single Work

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    In the previous work the author constructed a function of two variables taking algebraically independent values at any distinct points belonging to the cartesian product of the set of nonzero algebraic numbers and that of nonzero algebraic numbers inside the unit circle. In this paper the author constructs a balanced analogue of the two-variable function stated above, which is defined on a wider domain, the cartesian product of the set of complex numbers and that of complex numbers except the unit circle. This balanced function can also be regarded as a map defined on the set of orbits of the action of Klein four-group on the domain of definition of the balanced function and the latter map takes algebraically independent values at any distinct orbits represented by algebraic points.

  • Algebraic independence of the values of the Hecke-Mahler series and its derivatives at algebraic numbers

    Tanaka, Taka-aki, Tanuma, Yusuke

    Int. J. Number Theory (International Journal of Number Theory)  14 ( 9 ) 2369 - 2384 2018.10

    Research paper (scientific journal), Joint Work, Accepted,  ISSN  17930421

    　View Summary

    Hecke-Mahler series is the generating function of the sequence {[nω]} of the integral parts [nω] of the positive integer multiple nω for ω real. In this paper, we prove that the Hecke-Mahler series has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition, including the case where ω is in addition an algebraic integer. Using Mahler's method, we reduce the algebraic independence of the values of the Hecke-Mahler series to the linear independence of the Hecke-Mahler series themselves modulo the rational function field. For proving this linear independence, we observe the distribution of the sequence of points whose components consist of the fractional parts of the components of scalar
    multiple of a real vector.

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Papers, etc., Registered in KOARA 【 Display / hide 】

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Presentations 【 Display / hide 】

  • Algebraic independence of the values of a certain function invariant under the action of the dihedral group

    Tanaka, Taka-aki

    Analytic Number Theory and Related Topics (Kyoto) , 

    2024.10
    , 

    Oral presentation (general), Research Institute for Mathematical Sciences

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    Using a suitable linear recurrence of positive integers, we construct a certain function of three complex variables which are invariant under the action of the dihedral group on its domain of de nition. For such a function, we consider the map de ned on the set of orbits of the action of the dihedral group and its restriction to the set of orbits represented by algebraic points. Then we show that the restriction takes algebraically independent values at any distinct orbits represented by algebraic points.

  • 多変数Mahler関数の基礎と応用

    田中 孝明

    第31回 整数論サ マースクール (山形市) , 

    2024.09
    , 

    Oral presentation (invited, special), 山形大学

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    Mahler関数の定義から出発し, 値の代数的独立性の判定定理については, 複素数体上での最新結果および p進数体上と正標数の関数体上での結果も述べた.

  • On a certain function invariant under group action with remarkable algebraic independence properties

    Tanaka Taka-aki

    Diophantine Analysis and Related Fields 2024 (Kiryu) , 

    2024.03
    , 

    Oral presentation (general)

  • Mahler's method for algebraic independence of partial derivatives of certain series in several variables

    Haruki Ide, Taka-aki Tanaka

    Diophantine Analysis and Related Fields 2023 (Hirosaki) , 

    2023.03
    , 

    Oral presentation (general)

  • On power series generated by simpler sequences and having strong algebraic independence properties

    TANAKA TAKA-AKI

    Analytic Number Theory and Related Topics (Online via Zoom) , 

    2021.10
    , 

    Oral presentation (keynote), Research Institute for Mathematical Sciences

    　View Summary

    多項式でない冪級数であって、係数0が連続する部分の長さが無限大に発散するものを空隙級数とよぶ。空隙級数の複数の代数的数における値の間に生じ得る、代数的数係数の既約な多項式で表される関係式（代数的従属の関係式）は１次式に限られる。これは西岡等による先行研究により証明された。その中では、上述のような１次式となる関係式が成立するのは、係数0が連続する部分の長さの増大度が大きい場合は、比が１の冪根である代数的数における値の間に限られる、という定理が述べられている。その系として次のことが分かる。k!+k 次の項の係数のみが1で他の次数の係数は0である空隙級数は、単位円内の0以外の相異なる代数的数における値がすべて代数的独立となる。その理由は、k!+k が任意の正整数 N を法とする任意の剰余類に無限個分布することから、比が1の N乗根である代数的数における値の間には１次式で表される関係式さえも存在せず、従って、代数的数における値の間にいかなる代数的従属の関係式も存在しなくなるからである。
    一方、Loxton-van der Poortenによる先行研究では、固定された整数 d≧2 に対する d^k 次の項の係数のみが1である空隙級数に対しては、代数的従属の関係式は１次式に限られるものの、それらが必ずしも比が１の冪根ではない代数的数における値の間でも成立することが示されている。
    本講演では次の結果を解説する： d^k+k 次の項の係数のみが1で他の次数の係数は0である空隙級数は、単位円内の0以外の相異なる代数的数における値がすべて代数的独立となる、という k!+k の場合と同じ性質を有する。数列 d^k の構造が k! と比べて単純であることから、今回の結果の証明には k!+k の場合の証明とは本質的に異なる、降下法を用いた。

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • 完全代数的独立性の新しい展開

    2024.04
    -
    2028.03

    基盤研究(C), Principal investigator

  • 完全代数的独立性の拡張と高次元化

    2020.04
    -
    2024.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

  • 非アルキメデス付値体における完全代数的独立性

    2015.04
    -
    2019.03

    日本学術振興会, JSPS, No Setting

  • Ｍａｈｌｅｒ関数の特殊値による超越数の構造解明

    2010.04
    -
    2013.03

    日本学術振興会, JSPS, Principal investigator

  • ｐ進数体及び関数体上のＭａｈｌｅｒ関数の超越性と代数的独立性及びその応用

    2006.04
    -
    2009.03

    JSPS, 科学研究費補助金, Principal investigator

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Courses Taught 【 Display / hide 】

  • TOPICS IN ALGEBRA A

    2024

  • MATHEMATICS 2B

    2024

  • MATHEMATICS 2A

    2024

  • INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

    2024

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 2

    2024

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Courses Previously Taught 【 Display / hide 】

  • 代数学特論Ｃ

    慶應義塾大学大学院理工学研究科

    2018.04
    -
    2019.03

  • 代数学特論Ａ

    慶應義塾大学大学院理工学研究科

    2018.04
    -
    2019.03

  • 代数学第１同演習

    慶應義塾大学理工学部

    2018.04
    -
    2019.03

  • 代数学基礎同演習

    慶應義塾大学理工学部

    2018.04
    -
    2019.03

  • 関数論第１同演習

    慶應義塾大学理工学部

    2018.04
    -
    2019.03

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Social Activities 【 Display / hide 】

  • 日本気象予報士会

    1999.10
    -
    Present
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Memberships in Academic Societies 【 Display / hide 】

  • 日本数学会, 

    1995.01
    -
    Present
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