20次多项式中含三个循环根的Casas-Alvero猜想
The Casas-Alvero conjecture for three recycled roots in degree 20
AI总结：
本文围绕Casas-Alvero猜想展开，证明了两类次数的范式反例有限、p^r+1次范式反例具代数系数，且20次多项式中无含三个循环根的反例。
摘要：
Casas-Alvero猜想指出，若一个n次复一元多项式与其各阶导数都有公共根，则该多项式仅有一个根。本文给出三项结果：第一，次数为p^r+p^s或p^r+2p^s（p为素数，r、s为正整数）的范式可能反例数量有限；第二，次数为p^r+1的范式可能反例具有代数系数；最后，20次多项式中不存在含三个循环根的反例。
英文摘要：
The Casas-Alvero conjecture says that a degree $n$ complex univariate polynomial sharing a root with each of its derivative must have only one root. In this article we give three results. The first one, is that the number of possible counterexamples in normal form of degree $p^r+p^s$ or $p^r+2p^s$ is finite ($p$ prime, $r,s$ positive integers). The second result is that a possible counterexample in normal form of degree $p^r+1$ has algebraic coefficients and the final result is that in degree $20$ there are no counterexamples with three recycled roots.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
