Erdős Problems

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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 the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$ is, almost surely, \[\left(\frac{1}{2}+o(1)\right)n?\]

#522: [Er61,p.252]

analysis, polynomials, probability

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 $(\frac{2}{\pi}+o(1))\log n$.

Solved by Yakir [Ya21], who proved that almost all such polynomials have \[\frac{n}{2}+O(n^{9/10})\] many roots in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$.