论文标题
$ x^{3} + x^{2} y^{2} + y^{3} = k $的理性点
Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$
论文作者
论文摘要
我们研究了确定的问题，给定一个整数$ k $，是$ c_ {k}的有理解决方案：x^{3} z + x^{2} y^{2} + y^{3} z = kz^{4} $。对于$ k \ ne 0 $，曲线$ c_ {k} $具有$ 3 $，并且有从$ c_ {k} $到三个椭圆形曲线$ e_ {1，k} $，k} $，$ e_ {2，k} $，$ e_ {3，k} $。我们在$ C_ {K} $上明确确定这些椭圆曲线之一的合理点。我们讨论扩展结果涉及的挑战，以处理\ Mathbb {q} $中的所有$ k \。
We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the rational points on $C_{k}$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k \in \mathbb{Q}$.
