For a polynomial f∈C[X], let operatornamerootArgCount(f,I) be the number of roots of f, counted with multiplicity, whose arguments lie in I, where each argument is normalized to [0,2 pi). If f=a_0+…+a_dx^d, define M(f)=frac∑_i=0^d|a_i|√|a_0|,|a_d|. Does there exist a constant C∈R such that, for every polynomial f∈C[X] with a_0≠ 0 and every real numbers alpha,beta satisfying 0≤alpha≤beta≤2 pi, |operatornamerootArgCount(f,[alpha,beta])-fracbeta-alpha2 pideg(f)| ≤ C√|operatornamesupp(f)|log M(f)?

Erdős Problem #990 — Angular Distribution of Polynomial Roots

About 76 years old · traced to 1950

For a polynomial $f\in\mathbb{C}[X]$ , let $\operatorname{rootArgCount}(f,I)$ be the number of roots of $f$ , counted with multiplicity, whose arguments lie in $I$ , where each argument is normalized to $[0,2\pi)$ . If $f=a_0+\cdots+a_dx^d$ , define

M(f)=\frac{\sum_{i=0}^{d}|a_i|}{\sqrt{|a_0|\,|a_d|}}.

Does there exist a constant $C\in\mathbb{R}$ such that, for every polynomial $f\in\mathbb{C}[X]$ with $a_0\neq 0$ and every real numbers $\alpha,\beta$ satisfying $0\leq\alpha\leq\beta\leq2\pi$ ,

\left|\operatorname{rootArgCount}(f,[\alpha,\beta])-\frac{\beta-\alpha}{2\pi}\deg(f)\right| \leq C\sqrt{|\operatorname{supp}(f)|\log M(f)}?

References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

analysis

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Claimed solved

An explicit counterexample shows that the proposed bound is false, so the problem is settled negatively.

Erdős Problem #990 asks whether root angles of a sparse polynomial obey a discrepancy bound controlled by the number of nonzero terms rather than its degree. It is now disproved.

Known results

Erdős and Turán proved the analogous estimate with degree $d$ replacing the number of nonzero coefficients $n$ .
Hayman proved discrepancy at most $n-1$ and showed this is essentially sharp via $f(x)=(x^p-1)^{n-1}$ , though that example has large $M$ .

April 2026 counterexample

A construction gives, for every $N\geq 1$ , a polynomial with $N+2$ nonzero terms, $M<3$ , and a positive real root of multiplicity $N+1$ . An interval near angle zero therefore has discrepancy of order $N$ , contradicting every universal bound by $C\sqrt{n\log M}$ . The construction is presented in an arXiv preprint and is also attributed to an internal OpenAI model.

Current status (as of April 2026): The proposed bound is disproved by an explicit preprint construction; no remaining case of this problem is open.

An internal OpenAI modelOpenAIsolvedevidence

Sources

No solutions have been posted yet.