OPEN
This is open, and cannot be resolved with a finite computation.
Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$.
Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then\[\frac{R_n}{n/2}\to 1\]almost surely?
Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform $\pm 1$ values. Erdős and Offord
[EO56] showed that the number of real roots of a random degree $n$ polynomial with $\pm 1$ coefficients is $(\frac{2}{\pi}+o(1))\log n$.
There is some ambiguity whether Erdős intended the coefficients to be in $\{-1,1\}$ or $\{0,1\}$ - see the comments section.
A weaker version of this was solved by Yakir
[Ya21], who proved that\[\frac{R_n}{n/2}\to 1\]in probability. (This weaker claim was also asked by Erdős, and also appears in a book of Hayman
[Ha67].) More precisely,\[\lim_{n\to \infty} \mathbb{P}(\lvert R_n-n/2\rvert \geq n^{9/10}) =0.\]See also
[521].
View the LaTeX source
This page was last edited 06 December 2025.
Additional thanks to: Michal Bassan, Zachary Chase, and Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #522, https://www.erdosproblems.com/522, accessed 2026-01-16
