We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide...
The $S$-matrix conjecture was formulated by Sloane and Harwit in 1976, motivated by an $A$-optimal design problem arising in spectroscopy. It states that if $A$ is a nonsingular real $n\times n$ matrix whose entries lie in $[0,1]$, then $$\left\|A^{-1}\right\|_F\ge\frac{2n}{n+1}.$$ Moreover, equality holds if and only...
Very recently, Lech Mazur proved the celebrated Sendov conjecture, and Terence Tao subsequently distilled the main ideas of the proof in a blog post. In this paper, we establish a quantitative strengthening of Sendov's conjecture, namely the quadratic Tang--Zhang inequality. Let $p$ be a polynomial of degree $n\ge2$ wh...
Sendov's conjecture, first formulated in 1958, asserts that if a complex polynomial $f$ of degree $n \geq 2$ has all its zeros in the closed unit disk ${z \in \mathbb{C} : |z| \leq 1}$, then, for every zero $\lambda_0$ of $f$, there exists a critical point $\zeta$ of $f$ such that $|\zeta-\lambda_0| \leq 1$. The conjec...
Zhang Teng· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
