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 if $A$ is an $S$-matrix. In this paper, we prove the conjecture by combining a variational trace inequality in odd dimensions with a sharp range-constrained estimate for the Moore--Penrose inverse of a centered matrix in even dimensions.
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 $n\ge2$ whose zeros lie in the closed unit disk, and let $\zeta_1,\ldots,\zeta_{n-1}$ denote its critical points, counted with multiplicity. We prove that, for every zero $a$ of $p$, $$ \sum_{j=1}^{n-1}\frac{1}{|a-\zeta_j|^2}\ge n-1. $$ Moreover, equality holds if and only if $p(z)=c(z^n-\omega)$ for some $c\in\mathbb C\setminus\{0\}$ and $|\omega|=1$. We also provide a Lean 4 formalization of the main results.
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 conjecture was previously known to hold for polynomials of degree $n \leq 8$, as well as in several special higher-degree cases. In 2022, using compactness methods, balayage, and the argument principle, Tao~\cite{Tao22} proved that there exists an absolute constant $n_0$ such that Sendov's conjecture holds for all $n \geq n_0$. However, Tao's argument does not provide an explicit admissible value of $n_0$. In the present paper, we make Tao's result effective and prove that one may take $$ n_0 = 10^{200000}. $$
Zhang Teng· 0 citations
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