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| 研究生： |
胡家瑋 Hu, Chia-Wei |
|---|---|
| 論文名稱： |
位在一般位置的二重點上的插值問題 On the Interpolation Problem of Double Points in General Positions |
| 指導教授： |
卓士堯
Jow, Shin-Yao |
| 口試委員: |
陳俊成
Chen, Jiun-Cheng 鄭志豪 Teh, Jyh Haur |
| 學位類別： |
碩士 Master |
| 系所名稱： |
理學院 - 數學系 Department of Mathematics |
| 論文出版年： | 2020 |
| 畢業學年度： | 108 |
| 語文別： | 英文 |
| 論文頁數： | 40 |
| 中文關鍵詞： | 插值 、代數曲體 、割平面 、希爾伯特函數 |
| 外文關鍵詞： | Interpolation, Secant, Variety, Horace |
| 相關次數： | 點閱：1 下載：0 |
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代數幾何中，插值問題的目的在於尋找多項式，其零點依指定的重數通過給定的某些點。Alexander-Hirschowitz 定理給出了關於二重點上的插值問題的解答。在本篇論文中，我們將會探討由[Ch]與[BO]給出的Alexander-Hirschowitz定理的一個簡化後的證明，並以電腦驗證相關的結果。
The interpolation problem asks for the dimension of the linear system consisting of polynomials whose zero locus pass through a given collection of points with prescribed multiplicities. The Alexander-Hirschowitz theorem is the answer to the interpolation problem on general collections of double points. In this paper, we will review the simplified version of the proof of Alexander-Hirschowitz theorem given in [Ch] and [BO], explore the ideas behind the proof, and examine some of the results by computer.
[AH1] J. Alexander, A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math. 2000, 140, 303-325
[AH2] J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4 1995, n.2, 201-222
[BO] M.C. Brambilla, ; G. Ottaviani, On the Alexander-Hirschowitz theorem. J. Pure Appl. Algebra 2008, 212, 1229-1251
[B] Bernardi et al., The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition, Mathematics, 6.12, 2018, 314
[Ch] K. Chandler, A brief proof of a maximal rank theorem for generic double points in projective space, Trans, Amer, Math. Soc. 353, 2001, no. 5, 1907-1920