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

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be asubfield of $L$.  Consider $L$ as a vector space over $F$ and the associated projective spacethat is isomorphic to ${\mathrm{PG}}(n-1,q)$.  The properties of the projective mapping induced by $x\mapsto x^{-1}$ havebeen studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that theimage of any line is a normal rational curve in some subspace.  In this note a more detailed geometric description is achieved.  Consequences are found related to mixed partitions of the projective spaces;in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$,then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curvesof degree $2^k-1$.  For smaller $q$ the same construction gives partitions in $(q+1)$-tuples ofindependent points.

Geometry of the inversion in a finite field and partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves