{"ID":23021113,"CreatedAt":"2026-09-17T04:55:43.771560302Z","UpdatedAt":"2026-09-17T04:55:43.771560302Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19126","arxiv_id":"2609.19126","title":"Beyond Sendov's conjecture: the quadratic Tang--Zhang inequality","abstract":"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 $ζ_1,\\ldots,ζ_{n-1}$ denote its critical points, counted with multiplicity. We prove that, for every zero $a$ of $p$, $$ \\sum_{j=1}^{n-1}\\frac{1}{|a-ζ_j|^2}\\ge n-1. $$ Moreover, equality holds if and only if $p(z)=c(z^n-ω)$ for some $c\\in\\mathbb C\\setminus\\{0\\}$ and $|ω|=1$. We also provide a Lean 4 formalization of the main results.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2609.19126","url_pdf":"https://arxiv.org/pdf/2609.19126v1","authors":"[\"Teng Zhang\"]","published":"2026-09-16T17:46:45Z","proceeding":"math.CV","tasks":"[\"math.CV\"]","methods":"[]","has_code":false}