Irreducibility and hypertranscendence of non-linear difference equations

• Project/Area Number: 24840005
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Yamagata University
• Principal Investigator: NISHIOKA Seiji 山形大学, 理学部, 准教授 (10632226)
• Project Period (FY): 2012-08-31 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 離散パンルヴェ方程式 / 既約性 / 超越性 / 代数的差分方程式 / パンルヴェ方程式 / 分解可能拡大 / 差分代数 / 差分方程式

Research Abstract

There are special second-order algebraic difference equations called discrete Painleve equations. The equation called d-Painleve equation of type D7(1) is one of them. Its irreducibility in the sense of decomposable extension was proved. The irreducibility implies that the transcendental function solution cannot be built from rational functions by reiterating algebraic operations and the taking of solutions of linear difference equations or first-order algebraic difference equations. A standard form of difference Riccati equations for any transforming operator was also studied.

Research Products

• [Journal Article] Functions satisfying Poincare's multiplication formula (2014)
  • Author(s): Nishioka, S.
  • Journal Title: Osaka Journal of Mathematics Volume: 51
  • Peer Reviewed
• [Journal Article] Algebraic independence of solutions of first-order rational difference equations (2013)
  • Author(s): Nishioka S
  • Journal Title: Results in Mathematics Volume: Volume.64, Issue 3 Issue: 3-4 Pages: 423-433
  • DOI: 10.1007/s00025-013-0324-8
  • Peer Reviewed
• [Journal Article] Approximation of Poincare's new functions by rational functions (2013)
  • Author(s): Nishioka, S.
  • Journal Title: Journal of Difference Equations and Applications Volume: 19 Issue: 9 Pages: 1452-1462
  • DOI: 10.1080/10236198.2012.755963
  • Peer Reviewed
• [Journal Article] Solutions to a q-analog of Painleve III equation of type D7(1) (2013)
  • Author(s): Nakazono, N., Nishioka, S.
  • Journal Title: Funkcialaj Ekvacioj Volume: 56
  • Peer Reviewed
• [Presentation] 差分代数と超越数論 (2013)
  • Author(s): 西岡斉治
  • Organizer: Workshop on Galois point and related topics
  • Place of Presentation: 山形大学, 山形県
  • Year and Date: 2013-09-15
• [Presentation] 倍角公式と代数的独立性 (2013)
  • Author(s): 西岡斉治
  • Organizer: 2013函数方程式論サマーセミナー
  • Place of Presentation: リゾートホテル阿蘇いこいの村, 熊本県
  • Year and Date: 2013-08-09
• [Presentation] Approximation of Poincare's new functions by rational functions (2013)
  • Author(s): 西岡斉治
  • Organizer: Séminaire Equations fonctionnelles
  • Place of Presentation: Université de Strasbourg, フランス
  • Year and Date: 2013-02-12
• [Presentation] 差分リッカチ方程式の標準型 (2013)
  • Author(s): 西岡斉治
  • Organizer: ウィンターセミナー2013
  • Place of Presentation: KKR湯沢ゆきぐに, 新潟県
  • Year and Date: 2013-01-28
• [Presentation] 差分方程式から見た関数の初等性 (2012)
  • Author(s): 西岡斉治
  • Organizer: つくば微分ガロア理論セミナー
  • Place of Presentation: 筑波大学, 茨城県
  • Year and Date: 2012-12-13
• [Presentation] 差分代数と超越数論
  • Author(s): 西岡斉治
  • Organizer: Workshop on Galois point and related topics
  • Place of Presentation: 山形大学理学部
  • Invited
• [Presentation] 差分リッカチ方程式の標準型
  • Author(s): 西岡斉治
  • Organizer: ウィンターセミナー2013
  • Place of Presentation: KKR湯沢ゆきぐに（新潟県）