Resolution of singularities by using Newton polyhedra and its application to analysis

• Project/Area Number: 15K04932
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Kamimoto Joe 九州大学, 数理学研究院, 准教授 (90301374)
• Project Period (FY): 2015-04-01 – 2021-03-31
• Project Status: Completed (Fiscal Year 2020)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ニュートン多面体 / 特異点解消定理 / ベルグマン核 / ダンジェロの型 / 代数曲線 / 振動積分 / 局所ゼータ関数 / 特異点解消 / 接触位数 / ニュートン非退化 / 実超曲面 / 有限型 / 有限型領域 / ピーク関数

Outline of Final Research Achievements

By using Newton polyhedra, which is an important concept in singularity theory,
we developed a quantitative resolution of singularities theorem. Furthermore, we apply the resolution theorem to several complex variables and harmonic analysis and obtain many kinds of interesting results. To be more specific, we obtain strong results about the analytic continuation of local zeta functions, asymptotic analysis of oscillatory integrals and determination of D'Angelo's types.

Academic Significance and Societal Importance of the Research Achievements

代数や幾何における重要な成果をさらに発展させ応用することにより、今までに十分でなかった解析学における重要な問題について、多くの成果を得たこと。

Research Products

• [Journal Article] Newton polyhedra and order of contact on real hypersurfaces (2021)
  • Author(s): KAMIMOTO Joe
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 73 Issue: 1 Pages: 1-39
  • DOI: 10.2969/jmsj/80868086
  • NAID: 130007973712
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access
• [Journal Article] On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type (2020)
  • Author(s): Kamimoto Joe
  • Journal Title: The Journal of Geometric Analysis Volume: - Issue: 8 Pages: 1-17
  • DOI: 10.1007/s12220-020-00567-z
  • Peer Reviewed / Open Access
• [Journal Article] Meromorphy of local zeta functions in smooth model cases (2020)
  • Author(s): Kamimoto Joe、Nose Toshihiro
  • Journal Title: Journal of Functional Analysis Volume: 278 Issue: 6 Pages: 108408-108408
  • DOI: 10.1016/j.jfa.2019.108408
  • Peer Reviewed / Open Access
• [Journal Article] Newton polyhedra and order of contact on real hypersurfaces (2020)
  • Author(s): Kamimoto Joe
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 未定
  • Peer Reviewed / Open Access
• [Journal Article] A sufficient condition for equality of regular type and singular type of real hypersurfaces (2020)
  • Author(s): Kamimoto Joe
  • Journal Title: 数理解析研究所講究録 Volume: 未定
  • Open Access
• [Journal Article] Nonpolar singularities of local zeta functions in some smooth case (2019)
  • Author(s): Kamimoto Joe、Nose Toshihiro
  • Journal Title: Transactions of the American Mathematical Society Volume: 372 Issue: 1 Pages: 661-676
  • DOI: 10.1090/tran/7771
  • Peer Reviewed / Open Access
• [Journal Article] Nonpolar singularities of local zeta functions in some smooth case (2019)
  • Author(s): Joe Kamimoto and Toshihiro Nose
  • Journal Title: Trans. Amer. Math. Soc. Volume: 印刷中
  • Peer Reviewed / Open Access
• [Journal Article] A sufficient condition for equality of regular type and singular type fo real hypersurfaces (2019)
  • Author(s): Joe Kamimoto
  • Journal Title: 数理解析研究所講究録 Volume: 印刷中
  • Open Access
• [Journal Article] Non-polar singularities of local zeta functions in some smooth case (2018)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: Trans. Amer. Math. Soc. Volume: To appear
  • Peer Reviewed
• [Journal Article] Asymptotic limit of oscillatory integrals with certain smooth phases, (2017)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B63 Pages: 101-112
  • Peer Reviewed / Open Access
• [Journal Article] Asymptotic limit of oscillatory integrals with certain smooth phases (2017)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: RIMS K\^oky\^uroku Bessatsu Volume: 63 Pages: 101-112
  • Peer Reviewed / Open Access
• [Journal Article] Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals, (2016)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: J. Math. Soc. Univ. Tokyo Volume: 23 Pages: 425-485
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Newton polyhedra and weighted oscillatory integrals with smooth phases (2016)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: Trans. Amer. Math. Soc. Volume: 368 Pages: 5301-5361
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] On asymptotic expansions of oscillatory integrals with smooth phase in two dimensions (2016)
  • Author(s): J. Kamimoto and T. Nose
  • Journal Title: RIMS K\^oky\^uroku Bessatsu Volume: B57 Pages: 141-157
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Toric resolution of singularities in a certain class of $C^{\infty}$ functions and asymptotic analysis of oscillatory integrals (2016)
  • Author(s): Joe Kamimoto and Toshihiro Nose
  • Journal Title: J. Math. Soc. Univ. Tokyo Volume: 23
• [Journal Article] Newton polyhedra and weighted oscillatory integrals with smooth phases (2016)
  • Author(s): Joe Kamimoto and Toshihiro Nose
  • Journal Title: Trans. Amer. Math. Soc. Volume: 368
• [Journal Article] On asymptotic expansions of oscillatory integrals with smooth phase in two dimensions (2016)
  • Author(s): Joe Kamimoto and Toshihiro Nose
  • Journal Title: RIMS K\^oky\^uroku Bessatsu Volume: 未定
• [Journal Article] On meromorphic continuation of local zeta functions (2015)
  • Author(s): Joe Kamimoto and Toshihiro Nose
  • Journal Title: Proceedings of KSCV10, F. Bracci et al. (eds.) Volume: 144
• [Presentation] Asymptotic analysis of oscillatory integrals with degenerate phases (2021)
  • Author(s): Joe Kamimoto
  • Organizer: 偏微分方程式姫路研究集会
  • Int'l Joint Research / Invited
• [Presentation] C^∞ 平面曲線の特異点解消と局所ゼータ関数の有理型解析接続 (2020)
  • Author(s): 神本 丈
  • Organizer: 第63回函数論シンポジウム
  • Invited
• [Presentation] On holomorphic curves tangent to real hypersurfaces of infinite type (2020)
  • Author(s): 神本 丈
  • Organizer: 多変数関数論冬セミナー
  • Invited
• [Presentation] On meromorphy of local zeta functions (2020)
  • Author(s): Kamimoto Joe
  • Organizer: 研究集会「第15回代数・解析・幾何学セミナー」， 鹿児島大学
  • Invited
• [Presentation] 局所ゼータ関数の有理型解析接続可能領域について (2020)
  • Author(s): 神本丈, 野瀬敏洋
  • Organizer: 日本数学会2020年度春季総合分科会，日本大学
• [Presentation] 無限型擬凸領域のベルグマン核の境界挙動 (2019)
  • Author(s): 神本 丈
  • Organizer: 第54回函数論サマーセミナー，静岡県伊豆の国市
• [Presentation] 多変数関数論におけるニュートン多面体とその応用 (2019)
  • Author(s): 神本 丈
  • Organizer: 日本数学会2019年度秋季総合分科会，特別講演， 金沢大学
  • Invited
• [Presentation] Meromorphy of local zeta functions in smooth model cases (2019)
  • Author(s): Kamimoto Joe
  • Organizer: 研究集会「超局所解析と漸近解析」， 京大数理解析研究所
  • Invited
• [Presentation] ニュートン多面体と振動積分の漸近解析 I, II (2019)
  • Author(s): 神本 丈
  • Organizer: 筑波RCMS解析学シンポジウム
  • Invited
• [Presentation] Regular and singular orders of contact on real hypersurfaces. (2018)
  • Author(s): 神本 丈
  • Organizer: 第５３回函数論サマーセミナー
• [Presentation] Non-polar singularities of local zeta functions in smooth cases. (2018)
  • Author(s): 神本 丈
  • Organizer: 日本数学会2018年秋季総合分科会、岡山大学
• [Presentation] Regular and singular orders of contact on real hypersurfaces. (2018)
  • Author(s): 神本 丈
  • Organizer: 代数解析学の諸問題ー超局所解析及び漸近解析ー、京大数理解析
  • Invited
• [Presentation] ニュートン多面体と重みつき振動積分 (2018)
  • Author(s): 神本 丈
  • Organizer: 第２２８回 広島数理解析セミナー、広島大学
  • Invited
• [Presentation] Newton polyhedra and order of contact on real hypersurfaces. (2018)
  • Author(s): 神本 丈
  • Organizer: 複素解析幾何セミナー、東京大学
  • Invited
• [Presentation] ニュートン多面体を用いた特異点解消とその解析学への応用 (2018)
  • Author(s): 神本 丈
  • Organizer: 研究集会「接触構造、特異点、微分方程式及びその周辺」 金沢市，
  • Invited
• [Presentation] ベルグマン核の漸近解析， 2017年9月6日． (2017)
  • Author(s): 神本 丈
  • Organizer: 第52回函数論サマーセミナー，福岡県柳川市，
  • Invited
• [Presentation] Failure of meromorphy for local zeta functions, (2017)
  • Author(s): Joe Kamimoto
  • Organizer: RIMS 共同研究 (公開型)「超局所解析と漸近解析」 京大数理解析研
  • Int'l Joint Research / Invited
• [Presentation] On analytic continuation of local zeta functions (2016)
  • Author(s): Joe Kamimoto
  • Organizer: 研究集会「New development of microlocal analysis and singular perturbation theory」
  • Place of Presentation: 京大数理解析研
  • Year and Date: 2016-10-03
  • Int'l Joint Research / Invited
• [Presentation] Asymptotic analysis of oscillatory integrals and local zeta functions (2015)
  • Author(s): 神本 丈
  • Organizer: 研究集会「保存則をもつ偏微分方程式に対する解の正則性・特異性の研究」
  • Place of Presentation: 京大数理解析研究所
  • Year and Date: 2015-06-03
  • Invited