{"schema":"vela.problem-packet.v0.1","problem":990,"statement":"Let $f=a_0+\\cdots+a_dx^d\\in \\mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\\ldots,z_d$ with corresponding arguments $\\theta_1,\\ldots,\\theta_d\\in [0,2\\pi]$, then for all intervals $I\\subseteq [0,2\\pi]$\\[\\left\\lvert (\\# \\theta_i \\in I) - \\frac{\\lvert I\\rvert}{2\\pi}d\\right\\rvert \\ll \\left(n\\log M\\right)^{1/2},\\]where $n$ is the number of non-zero coefficients of $f$ and\\[M=\\frac{\\lvert a_0\\rvert+\\cdots +\\lvert a_d\\rvert}{(\\lvert a_0\\rvert\\lvert a_d\\rvert)^{1/2}}.\\]","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}