论文标题
一个足够的条件使复杂多项式仅具有简单的零和Hutchinson定理的真实多项式的类似物
A sufficient condition for a complex polynomial to have only simple zeros and an analog of Hutchinson's theorem for real polynomials
论文作者
论文摘要
我们找到常数$ b _ {\ infty} $（$ b _ {\ infty} \大约4.81058280 $），以便如果一个复杂的多项式或整个函数$ f（z）= \ sum_ {k = 0}^ 0}^Ωa_a_kz^ k^ k^ k，$ $ $ $ $ $ $ $ $ $ cup， \ {\ infty \}，具有非零系数的$满足条件$ \ left | \ frac {a_k^2} {a_ {a_ {k-1} a_ {k+1}}} \ right | > b _ {\ infty} $ for ALL $ k = 1，2，\ ldots，ω-1，$，然后$ f $的所有零都很简单。我们证明了上述声明中常数$ b _ {\ infty} $是最小的。我们还获得了Hutchinson定理的类似物，用于多项式或具有实际非零系数的整个函数。
We find the constant $b_{\infty}$ ($b_{\infty} \approx 4.81058280$) such that if a complex polynomial or entire function $f(z) = \sum_{k=0}^ ωa_k z^k, $ $ω\in \{2, 3, 4, \ldots \} \cup \{\infty\},$ with nonzero coefficients satisfy the conditions $\left|\frac{a_k^2}{a_{k-1} a_{k+1}}\right| >b_{\infty} $ for all $k =1, 2, \ldots, ω-1,$ then all the zeros of $f$ are simple. We show that the constant $b_{\infty}$ in the statement above is the smallest possible. We also obtain an analog of Hutchinson's theorem for polynomials or entire functions with real nonzero coefficients.
