TITLE: Is there a better way to find the polynomial equation for this curve?
QUESTION [10 upvotes]: Consider the curve in $\mathbb{R}^2$ defined by the equation
$$
x^{1/3} + y^{1/3} + (xy)^{1/3} = 1,
$$
where $x^{1/3}$ denotes the real cube root of $x$, etc.
Since the equation above involves only algebraic operations, this curve ought to be a branch of an algebraic curve, i.e. a curve defined by a polynomial equation involving $x$ and $y$.  I know of only one algorithm to find this equation:

Compute the minimum polynomial of $x^{1/3} + y^{1/3} + (xy)^{1/3} - 1$ over $\mathbb{Q}(x,y)$.  Since $\mathbb{Q}(x^{1/3},y^{1/3})$ has degree $9$ over $\mathbb{Q}(x,y)$, the minimum polynomial $p \in \mathbb{Q}(x,y)[t]$ presumably has degree $9$, so this will involve computing and then row reducing a $9\times 10$ matrix over $\mathbb{Q}(x,y)$.
The "constant term" of $p$ will be a rational function in $x$ and $y$ whose numerator is a polynomial $q(x,y)$.  Then the given curve is a branch of the algebraic curve defined by the equation $q(x,y)=0$.

Is there some better way to do this?

REPLY [2 votes]: Two automated approaches.  With Gröbner basis: see the first entry of this result.  With resultants: first compute this and then this.