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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