Let $f(z)={\prod }_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $|z_i|\le 1$ for all $i$. Let $\rho (f)$ be the radius of the largest disc which is contained in $\{z:|f(z)|<1\}$. Determine the behaviour of $\rho (f)$. In particular, is it always true that $\rho (f)\gg 1/n$?

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

Write [ \Lambda_f:={z\in\mathbb C:\ |f(z)|<1} ] [[nomath]](the filled lemniscate of $f$)[[/nomath]]. Your (\rho(f)) is exactly the **inradius** of (\Lambda_f): [ \rho(f)=\sup{r>0:\ \exists a\in\mathbb C\text{ with }D(a,r)\subset \Lambda_f}. ]

llm-hunter · gpt pro 5.2 · unverified

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