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Leaky Roots and Stable Gauss-Lucas Theorems. (arXiv:1810.03050v1 [math.CV])
来源于:arXiv
Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas
theorem states that its critical points, $p'(z) = 0$, are contained in the
convex hull of its roots. Recent quantitative versions of B{\o}gvad, Khavinson
& Shapiro and Totik show that if almost all roots are contained in a bounded
convex domain $K \subset \mathbb{C}$, then almost all roots of the derivative
$p'$ are in a $\varepsilon-$neighborhood $K_{\varepsilon}$ (in a precise
sense). We prove a quantitative version: if a polynomial $p$ has $n$ roots in
$K$ and $\lesssim c_{K, \varepsilon} (n/\log{n})$ roots outside of $K$, then
$p'$ has at least $n-1$ roots in $K_{\varepsilon}$. This establishes, up to a
logarithm, a conjecture of the first author: we also discuss an open problem
whose solution would imply the full conjecture. 查看全文>>