The number of roots of f, counted with multiplicity, whose argument lies in I. The argument is normalised to [0, 2π).

For $f = a_0 + \cdots + a_dx^d$, the quantity $M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}$.

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 $$ M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}. $$

An internal OpenAI model (see [APSSV26b]) has disproved the conjecture, constructing, for every $n\geq 1$, a polynomial $f$ with $n$ non-zero coefficients such that $M<3$ and with a positive real zero of multiplicity $n-1$, whence letting $I=[0,c/d]$ for a suitably small $c>0$, $$ \left\lvert (# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \geq n-1. $$

Erdős and Turán [ErTu50] proved such an upper bound with $n$ replaced by $d$.

Hayman [Ha72b] proved $$ \left\lvert (# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \leq n-1, $$ and noted this is essentially sharp since $f(x)=(x^{p}-1)^{n-1}$ has $n$ non-zero coefficients and has $1$ as a positive real zero of multiplicity $n-1$ (although for this $f$ the parameter $M$ becomes very large).

An internal OpenAI model (see [APSSV26b]) has disproved the conjecture, constructing, for every $n\geq 1$, a polynomial $f$ with $n$ non-zero coefficients such that $M<3$ and with a positive real zero of multiplicity $n-1$.