Let $p$ be a complex polynomial of degree $n\ge2$, and let $\sigma_2(p)$ be the root mean square distance of its zeros from their centroid, including multiplicities. We prove that every zero of $p$ lies within distance $\sigma_2(p)$ of a critical point. This proves Borcea's quadratic conjecture and, in particular, give...
Let $(a_n)_{n\geq0}$ be a sequence of complex numbers, and define \[ b_n=\sum_{k=0}^n \binom{n}{k} a_k, \qquad c_n=\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}a_k. \] Let $\beta>1$, put $\alpha=\sqrt{\beta^2-1}$, and suppose that $b_n,c_n=O(\beta^n)$. Mashreghi and Ransford proved that \[ \limsup_{n\to\infty}\frac{|a_n|}{\alpha...
Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserv...
Ludovick Bouthat, Dominique Guillot· 0 citations
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