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

The point-plane incidence theorem states that the number of incidencesbetween $n$ points and $m\geq n$ planes in the projective three-space over afield $F$, is $$O\left(m\sqrt{n}+ m k\right),$$ where $k$ is the maximum numberof collinear points, with the extra condition $n< p^2$ if $F$ hascharacteristic $p>0$. This theorem also underlies a state-of-the-artSzemer\'edi-Trotter type bound for point-line incidences in $F^2$, due toStevens and de Zeeuw.  This review focuses on some recent, as well as new, applications of thesebounds that lead to progress in several open geometric questions in $F^d$, for$d=2,3,4$. These are the problem of the minimum number of distinct nonzerovalues of a non-degenerate bilinear form on a point set in $d=2$, the analogueof the Erd\H os distinct distance problem in $d=2,3$ and additive energyestimates for sets, supported on a paraboloid and sphere in $d=3,4$. It avoidsdiscussing sum-product type problems (corresponding to the special case ofincidences with Cartesian products), which have lately received more attention.

Point-plane incidences and some applications in positive characteristic