Necessary and sufficient coefficient conditions for polynomial root anti-concentration

Characterize natural necessary and sufficient conditions on coefficient-law classes for the polynomial root-hitting constant C_D to be finite and, up to polynomial factors, bounded polynomially in the degree d and coefficient radius R, including the small-R regime.

Background

For random monic polynomials with coefficients in [-R,R], the root-hitting constant measures the maximum probability per interval length that a polynomial has a root in a given interval. The source problem asks for a natural characterization of coefficient distributions that ensures finite or polynomially bounded anti-concentration.

The paper proves a polynomial upper bound under endpoint conditional-density mean caps, but this is only a sufficient condition. It explicitly leaves open a necessary-and-sufficient coefficient-side characterization for general coefficient-law classes and all R>0.

References

The first subproblem of \citet{balcan2026online} asks: \textit{under what natural necessary and sufficient conditions on $\mathcal D$ is $C_{\mathcal D}$ finite, and under what conditions is it bounded by a polynomial in $d$ and $R$?}

— VALG: An Agentic System for ML Theory Research  (2608.13060 - Zhang et al., 13 Aug 2026) in Section 4, Subsection “Online Optimization of Piecewise-Lipschitz Functions with Applications to Data-Driven Algorithm Design”, Subproblem 1: Polynomial Boundaries

The second subproblem of \citet{balcan2026online} asks: \textit{what normalization of $F$, analogous to fixing the leading coefficient of a polynomial to one, guarantees that $C{\mathrm{Pf}_{\mathcal D}$ is finite?}

— VALG: An Agentic System for ML Theory Research  (2608.13060 - Zhang et al., 13 Aug 2026) in Section 4, Subsection “Online Optimization of Piecewise-Lipschitz Functions with Applications to Data-Driven Algorithm Design”, Subproblem 2: Pfaffian Boundaries