English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Conditional on the extended Riemann hypothesis, we show that with highprobability, the characteristic polynomial of a random symmetric $\{\pm1\}$-matrix is irreducible. This addresses a question raised by Eberhard inrecent work. The main innovation in our work is establishing sharp estimatesregarding the rank distribution of symmetric random $\{\pm 1\}$-matrices over$\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$. Previously, suchestimates were available only for $p = o(n^{1/8})$. At the heart of our proofis a way to combine multiple inverse Littlewood--Offord-type results to controlthe contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$with anticoncentration at least $1/p + \Omega(1/p^2)$. Previously, inverseLittlewood--Offord-type results only allowed control over vectors withanticoncentration at least $C/p$ for some large constant $C > 1$.

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial