Problem 990

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}}.

Source: erdosproblems.com/990

Analysis

Wiki page

Status

SolvedDisproved

An accepted solution exists. The statement is false.

DISPROVED (LEAN). The site credits the construction in [APSSV26b], whose authors attribute the proof to an internal OpenAI model; its Lean marker refers to third-party formalizations that this corpus has not built. The standing derives from the claim page (Alexeev Putterman Sawhney Sellke Valiant, 2026).

Problem 990

Status

Proof claims

Discussion