Casas-Alvero猜想的假设反例约束
Constraints on hypothetical counterexamples to the Casas-Alvero conjecture
摘要：
Casas-Alvero猜想指出：如果一个复一元多项式与它的每个导数有一个公共根，那么它有一个唯一的根。我们证明了假设反例必须至少有5个不同的根。第一种情况下，猜想是未知的是在度12。我们研究的情况下，度12，更一般的程度p+1，其中p是一个素数。虽然我们没有接近解决12度的猜想，但我们提出了反例必须满足的几个进一步的约束。
英文摘要：
The Casas-Alvero conjecture states: if a complex univariate polynomial has a common root with each of its derivatives, then it has a unique root. We show that hypothetical counterexamples must have at least 5 different roots. The first case where the conjecture is not known is in degree 12. We study the case of degree 12, and more generally degree p+1, where p is a prime number. While we don't come closing to solving the conjecture in degree 12, we present several further constraints that counterexamples would have to satisfy.
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