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

Xiaoan Lang, Jeremy Rouse

We study the problem of determining, given an integer $k$, the rational solutions to $C_k:x^{3}z+x^2{y}^2+y^{3}z=k{z}^4$. For $k\ne 0$, the curve $C_k$ has genus $3…