无限次Casas-Alvero猜想
The Casas-Alvero conjecture for infinitely many degrees
摘要：
在特征为零的域上，很明显，形式为（X-a）^d的多项式与其d-1个一阶导数中的每一个具有非平凡的公共因子。卡萨斯-阿尔韦罗（Casas-Alvero）已经证明了这一点。到目前为止，只有一些计算验证小度d。本文在多项式的次数是素数的幂或其两倍幂的情况下证明了这个猜想。此外，对于每个正特征p，我们给出了一个例子，一个多项式的次数d，这是不是一个d次幂，但有一个公共因子，其第一个d-1导数。这表明，特征零的假设对于逆命题成立是必不可少的。
英文摘要：
Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been some computational verifications for small degrees d. In this paper the conjecture is proved in the case where the degree of the polynomial is a power of a prime number, or twice such a power. Moreover, for each positive characteristic p, we give an example of a polynomial of degree d which is not a dth power but which has a common factor with each of its first d-1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.
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