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Erdős #522 · analysis / polynomials

Let f(z)=∑0≤k≤nϵkzkf(z)=\sum_{0\leq k\leq n} \epsilon_k z^k be a random polynomial, where ϵk∈{−1,1}\epsilon_k\in \{-1,1\} independently uniformly at random for 0≤k≤n0\leq k\leq n.

Is it true that, if RnR_n is the number of roots of f(z)f(z) in…

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let f(z)=∑0≤k≤nϵkzkf(z)=\sum_{0\leq k\leq n} \epsilon_k z^k be a random polynomial, where ϵk∈{−1,1}\epsilon_k\in \{-1,1\} independently uniformly at random for 0≤k≤n0\leq k\leq n.

Is it true that, if RnR_n is the number of roots of f(z)f(z) in {z∈C:∣z∣≤1}\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}, then

Rnn/2→1\frac{R_n}{n/2}\to 1

almost surely?

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Formal statement

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Background

Random polynomials with independently identically distributed coefficients are sometimes called Kac polynomials - this problem considers the case of Rademacher coefficients, i.e. independent uniform ±1\pm 1 values. Erdős and Offord [EO56] showed that the number of real roots of a random degree nn polynomial with ±1\pm 1 coefficients is (2π+o(1))log⁡n(\frac{2}{\pi}+o(1))\log n.

There is some ambiguity whether Erdős intended the coefficients to be in {−1,1}\{-1,1\} or {0,1}\{0,1\} - see the comments section.

A weaker version of this was solved by Yakir [Ya21], who proved that

Rnn/2→1\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→∞P(∣Rn−n/2∣≥n9/10)=0.\lim_{n\to \infty} \mathbb{P}(\lvert R_n-n/2\rvert \geq n^{9/10}) =0.

See also [521].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #522 link
  2. [2]Upstream Lean 4 statement for Erdős #522 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 522 link

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