论文标题
Sendov的猜想是足够高度多项式的
Sendov's conjecture for sufficiently high degree polynomials
论文作者
论文摘要
Sendov的猜想断言，如果一个复杂的多项式$ f $ of度数$ n \ geq 2 $具有其所有零零，则在封闭的单位磁盘中$ \ {z：| z | | \ leq 1 \} $，然后对于每个这样的零$λ_0$，封闭的单位磁盘$ \ {z：| z-λ_0| \ leq 1 \} $。该猜想以$ n <9 $而闻名，但仅适用于$ n $的部分结果。我们证明存在一个常数的$ n_0 $，因此Sendov的猜想以$ n \ geq n_0 $所持有。对于$λ_0$，距离原点和单位圆圈，我们可以诉诸于先前的Dégot和Chalebgwa的工作；对于$λ_0$，我们在单位圆圈附近，我们完善了先前的米勒（Miller）论点（当$λ_0$非常接近单位圈时，还调用了chijiwa的结果）；对于原点附近的$λ_0$，我们使用紧凑型方法，balayage和参数原理引入了一个新的参数。
Sendov's conjecture asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all of its zeroes in closed unit disk $\{ z: |z| \leq 1 \}$, then for each such zero $λ_0$ there is a zero of the derivative $f'$ in the closed unit disk $\{ z: |z-λ_0| \leq 1 \}$. This conjecture is known for $n < 9$, but only partial results are available for higher $n$. We show that there exists a constant $n_0$ such that Sendov's conjecture holds for $n \geq n_0$. For $λ_0$ away from the origin and the unit circle we can appeal to the prior work of Dégot and Chalebgwa; for $λ_0$ near the unit circle we refine a previous argument of Miller (and also invoke results of Chijiwa when $λ_0$ is extremely close to the unit circle); and for $λ_0$ near the origin we introduce a new argument using compactness methods, balayage, and the argument principle.
