Geometric structure of mixed non-degenerate singularity

• Project/Area Number: 25400098
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Oka Mutsuo 東京理科大学, 理学部, 教授 (40011697)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 混合多項式 / ミルナー束 / ニュートン境界 / ミルナーファイバー / レンズ方程式 / 混合超曲面 / 混合特異点 / 極擬斉次多項式 / ラディアル擬斉次多項式 / 非退化性 / 基本群

Outline of Final Research Achievements

We clarrified the structure of Newton non-degenerate mixed singularities. In particular, we gave a formula for the zeta function of mixed function with strongly polar weighted face functions. We also showed the existence of the contact structure on such a link. We also showed the relation of the moduli space of strongly mixed homogeneous polynomials of two variables and the moduli of mixed polynomials of one variables and the range of possible number of roots.It is also shown that the Lens equation studied in astro-physics has strongly related to our moduli problem.

Research Products

• [Journal Article] Topology of strongly polar weighted homogeneous links (2015)
  • Author(s): V.Blanloeil and M. Oka
  • Journal Title: SUT. J. MATH. Volume: 51 Pages: 119-128
  • NAID: 120006397913
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] On Milnor fibrations of mixed functions, af-condition and boundary stability (2015)
  • Author(s): M. Oka
  • Journal Title: Kodai J. Math. Volume: 38 Pages: 581-603
  • NAID: 130005475928
  • Peer Reviewed / Open Access
• [Journal Article] Classification of the fundamental groups of join type curves of degree 7 (2015)
  • Author(s): C.Eyral and M. Oka
  • Journal Title: J. Math.Soc. Japan Volume: 67 Pages: 663-698
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Mixed function of strongly polar weighted homogeneus face type (2015)
  • Author(s): Mutsuo Oka
  • Journal Title: Advanced Study of Mathematics Volume: 66 Pages: 173-202
  • Peer Reviewed
• [Journal Article] Contact structure on mixed links (2014)
  • Author(s): Mutsuo Oka
  • Journal Title: Vietnam J. Math. Volume: 42 Pages: 249-271
  • Peer Reviewed
• [Journal Article] Fundamental groups of join-type curves-achievements and perspectives (2014)
  • Author(s): C.Eyral and M. Oka
  • Journal Title: Proc. Japan Acad. Ser. A Volume: 90 Pages: 43-47
  • NAID: 40019983641
  • Peer Reviewed
• [Journal Article] On the fundamental groups of non-generic R-Join-type curves (2014)
  • Author(s): C.Eyral and M.Oka
  • Journal Title: Springer Proceeding in Mathematics and Statistics Volume: 96 Pages: 137-157
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Fundamental groups of join-type curves―achievements and perspectives. (2014)
  • Author(s): C.Eyral and M. Oka
  • Journal Title: Proc. Japan. Acad. Volume: 90 Pages: 43-47
  • NAID: 40019983641
  • Peer Reviewed
• [Journal Article] Fundamental groups of join-type sextics via dessins d'enfants (2013)
  • Author(s): C. Eyral and M. Oka
  • Journal Title: Proc. London Math. Soc. Volume: 107 Pages: 76-120
  • Peer Reviewed
• [Presentation] On the Milnor fibration of mixed homogeneous functions and af condition (2015)
  • Author(s): 岡 睦雄
  • Organizer: 30 Coloquio Brasileiro de Mathematica
  • Place of Presentation: ブラジル、リオデジャネイロ、IMPA
  • Year and Date: 2015-07-29
• [Presentation] On the generalized Lens equation and Milnor fibration (2015)
  • Author(s): 岡 睦雄
  • Organizer: Mexico-Brasil Singularity conference, Salvador
  • Place of Presentation: ブラジル、サルバドール大学
  • Year and Date: 2015-07-16
• [Presentation] On the generalizd Lens equations and its applications (2015)
  • Author(s): 岡睦雄
  • Organizer: Hanoi Mathematics Institute Seminar
  • Place of Presentation: ベトナム、ハノイ数学研究所
  • Year and Date: 2015-04-09
• [Presentation] On the fundamental groups of non-generic R Join-type curves (2015)
  • Author(s): Mutsuo OKA
  • Organizer: Branched Coverings,Degenerations, and related Topics
  • Place of Presentation: 首都大学東京
  • Year and Date: 2015-03-07
• [Presentation] On non-isolated mixed singularities and its Minor fibration (2014)
  • Author(s): M. Oka
  • Organizer: Workshop on Singularities in Geometry,topology, foliation's and dynamics
  • Place of Presentation: Universite Autonoma de Yacatan
  • Year and Date: 2014-12-16
  • Invited
• [Presentation] Introduction to mixed hypersurface singularities (2014)
  • Author(s): M. Oka
  • Organizer: CIMPA school on singularities
  • Place of Presentation: Cuernavaca大学
  • Year and Date: 2014-11-24 – 2014-11-28
  • Invited
• [Presentation] On non-degenerate mixed functions (2014)
  • Author(s): M. Oka
  • Organizer: International Workshop on Real and Complex Singularities
  • Place of Presentation: Sao Carlos 大学
  • Year and Date: 2014-08-01
  • Invited
• [Presentation] Introduction to the fundamental group of plane curves (2013)
  • Author(s): MutsuoOka
  • Organizer: Hanoi CIMPA school of Mathematics
  • Place of Presentation: ハノイ数学研究所
  • Invited
• [Presentation] On the fundamental group of non-generic R-join type curves
  • Author(s): Mutsuo Oka
  • Organizer: "Experimental and Theoretical Methods in Algebra, Geometry and Topology
  • Place of Presentation: Romania、Eforie Nord
  • Invited
• [Presentation] On the geometry of non-degenerate mixed hypersurface
  • Author(s): Mutsuo Oka
  • Organizer: Geometric singularity theory
  • Place of Presentation: Banach センター、ワルシャワ
  • Invited
• [Presentation] On the fundamental group of Join type plane curves
  • Author(s): Mutsuo Oka
  • Organizer: Japanese Turkish Geometry Meeting
  • Place of Presentation: Galatasaray大学、Istanbul
  • Invited