Let $f=a_0+\cdots +a_d{x}^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\dots ,z_d$ with corresponding arguments ${\theta }_1,\dots ,{\theta }_d\in [0,2\pi ]$, then for all intervals $I\subseteq [0,2\pi ]$$$\left|(\#{\theta }_i\in I)-\frac{|I|}{2\pi }d\right|\ll {\left(n\log M\right)}^{1/2},$$where $n$ is the number of non-zero coefficients of $f$ and$$M=\frac{|a_0|+\cdots +|a_d|}{(|a_0||a_d|{)}^{1/2}}.$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you wrote is **exactly the classical Erdős–Turán angular discrepancy bound**, except that in the known theorem the parameter under the square root is the **degree** $d$, not the **number $n$ of nonzero coefficients**.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.