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 study the problem of determining, given an integer $k$, the rationalsolutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$,the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to threeelliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine therational points on $C_{k}$ under the assumption that one of these ellipticcurves has rank zero. We discuss the challenges involved in extending ourresult to handle all $k \in \mathbb{Q}$.

Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$