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研究生： 施政邦
Shih, Cheng-Pang
論文名稱： Casas-Alvero猜想
On the Casas-Alvero Conjecture
指導教授： 卓士堯
Jow, Shin-Yao
口試委員: 陳俊成
Chen, Jiun-Cheng
陳正傑
Chen, Jheng-Jie
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： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 35
中文關鍵詞： 單變數多項式 、Casas-Alvero猜想 、代數幾何
外文關鍵詞： Univariate Polynomials, Casas-Alvero Conjecture, Algebraic Geometry
相關次數： 點閱：75  下載：3 
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  •  上一筆

    • Casas-Alvero 猜想描述的內容為：在一個特徵零的體上的一元多項式環
    中，只有一個線性多項式的完全次方才能同時跟他的每階微分都有公根。
    我們在本文中蒐集了許多關於證明此猜想過程中的結果，同時加入我們自
    己的觀點，並提供了在我們這樣的觀點下能得到的部分結果。

    The Casas-Alvero conjecture states that the only polynomial over a field
    of characteristic zero in one variable that has a common root with each of
    its derivative is a power of a linear polynomial. We collect various results
    towards proving this conjecture, and add our point of view as well as partial
    results from such a view.

    1 Introduction 4 2 Conventions, Definitions, and the Conjecture 5 2.1 Background Material . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Graf von Bothmer, Labs, Schicho, van de Woestijne’s Paper 7 3.1 Two preliminary lemmas involving binomial coefficients . . . . 7 3.2 Using aforementioned lemmas to prove the conjecture under certain conditions in characteristic p . . . . . . . . . . . . . . 17 3.3 Extension to characteristic 0 from p . . . . . . . . . . . . . . . 22 4 Draisma, de Jong’s Paper 24 4.1 p-adic valuation . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.2 The Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5 Castryck, Laterveer, Ouna¨ıes’s Paper 28 6 Our Attempt 31 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.2 Partial Results . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7 Conclusion 34 References 35 3

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    [DJ11] Jan Draisma and Johan P. de Jong. “On the Casas-Alvero conjecture”.
    In: Eur. Math. Soc. Newsl 80 (2011), pp. 29–33.
    [CLO12] Wouter Castryck, Robert Laterveer, and Myriam Ouna¨ıes. Constraints
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    verification in degree 12. 2012. doi: 10.48550/ARXIV.1208.5404.
    url: https://arxiv.org/abs/1208.5404.

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