Skip to content

Erdős problem 990

Let f=a0+⋯+adxd∈C[x]f=a_0+\cdots+a_dx^d\in \mathbb{C}[x] be a polynomial. Is it true that, if ff has roots z1,…,zdz_1,\ldots,z_d with corresponding arguments θ1,…,θd∈[0,2π]\theta_1,\ldots,\theta_d\in [0,2\pi], then for all intervals I⊆[0,2π]I\subseteq [0,2\pi] ∣(#θi∈I)−∣I∣2πd∣≪(nlog⁡M)1/2, \left\lvert (\# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \ll \left(n\log M\right)^{1/2}, where nn is the number of non-zero coefficients of ff and M=∣a0∣+⋯+∣ad∣(∣a0∣∣ad∣)1/2. M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}.

How the state got here

1 of 6 recorded
  1. Prior stateNone retained
  2. WorkNo submitted work retained
  3. VerificationNo check retained
  4. DecisionNo Repository has decided
  5. StandingA Result stands here
  6. Next gapNot recorded

Result history

Published changes, performers, checks, and later corrections.

No result history yet
No proposed change is retained for this Problem, so there is nothing to show a decision on.

Correction history

No correction history

Technical detailsExact roots, source, and retained record identifiers

Exact provenance

Problem row
sha256:5dc4253322773a5595537387d098434f8c79eb26ab1d2123c96de61ca0b0bf64
Metadata
sha256:0a0c9dece6a58e5c0efb0c56defb46e92b2f7d4856f3dcf9ceb9751b657b7fc4
Observation
sha256:9c8326e865b6017b103af7611c852a06776d81964979b8ab5a1b800cd550dff6
Content
sha256:464e019229ef8b5687e6a7e7efd72f3d507816552c9cbcc5c265002836bd01dd
Repository
sha256:a956b84c437202e5a02cc9e036a621bd14a302b34a75758115730bdbb77c52a4
Projection
sha256:a4a96441668a9f8b33afbcb3f696a2639f5fae145f36ddc063d7f67d5fb87ce8
Source commit
36c6d0fef71a3fec84d9dcd5eeca4e22e378f7cb