google-deepmind / google-deepmind/formal-conjectures
Erdős Problem 1039
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Description
What is the conjecture
https://www.erdosproblems.com/1039
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[x]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$.
Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$?
Status: open
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Contributor guide
First steps
- Read the whole issue, then the project's contributing guide.
- Comment on the issue to say you are picking it up — it saves two people doing the same work.
- Fork the repository and make your change on a branch.
- Open a pull request that references the issue number.
Research direction
Start by reading the Erdős Problems 1039 page and the formal-conjectures repository's existing formalized conjectures to identify the project's conventions for encoding polynomial statements and open problems. The work is done when Problem 1039 is added as a valid Lean formalization that captures the stated definition of ρ(f) and the question about its lower bound.
Written by the indexing model from the issue text.
Assessment
- Domain
- tooling
- Issue type
- Feature
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Active
- Clarity
- Mostly clear
- Newbie friendliness
- 45/100