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/

Global: Zendy Plus annual

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

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove the existence of complex polynomials $p(z)$ of degree $n$ and $q(z)$of degree $m<n$ such that the harmonic polynomial $ p(z) + \overline{q(z)}$ hasat least $\lceil n \sqrt{m} \rceil$ many zeros. This provides an array of newcounterexamples to Wilmshurst's conjecture that the maximum valence of harmonicpolynomials $p(z)+\overline{q(z)}$ taken over polynomials $p$ of degree $n$ and$q$ of degree $m$ is $m(m-1)+3n-2$. More broadly, these examples show thatthere does not exist a linear (in $n$) bound on the valence with a uniform (in$m$) growth rate. The proof of this result uses a probabilistic technique basedon estimating the average number of zeros of a certain family of randomharmonic polynomials.

The valence of harmonic polynomials viewed through the probabilistic lens