Finding relations of variables

Question

Suppose that

\begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*}

Find a polynomial $P(x, y, z)=0$ relating $x$, $y$ and $z$.

In general, if we are given \begin{align*} x_1&=P_1(t_1, \cdots, t_m)\\ \vdots&\\ x_n&=P_n(t_1, \cdots, t_m) \end{align*} where $P_1, \cdots, P_n$ are polynomials, is there a general way we can find polynomial relations of $x_1, \cdots, x_n$? I know that when $n=2$ and $m=1$, a polynomial relation is given by the resultant of $P_1(t_1)-x_1$ and $P_2(t_1)-x_2$.

Answer 1

As suggested in the comments, Cox, Little, and O'Shea have written about how to solve the implicitization problem, but in their first book Ideals, Varieties, and Algorithms. See my answer here for a reference to the result used below. (I think we could alternatively use multivariable resultants rather than Gröbner bases.)

Let $u = s^{-1}$ and $v = t^{-1}$. We form the ideal \begin{align*} I = (su - 1, tv - 1, x &- (t + v + t^2 s + v^2 u + t u + v s - 6),\\ y &- (t + v^2 + t s + u - 4), z - (v + t^2 + v u + s - 4)) \end{align*} in $\mathbb{Q}[s,t,u,v,x,y,z])$, and compute a Gröbner basis $G$ with respect to the lexicographical order. The last entry of $G$ is a Gröbner basis for the elimination ideal $I \cap \mathbb{Q}[x,y,z]$, and is the polynomial $P(x,y,z)$ you're looking for. Computing using SageMath, we find \begin{align*} P(x,y,z) = x^{5} + 2 x^{4} y + 2 x^{4} z + 50 x^{4} - x^{3} y^{2} z - 7 x^{3} y^{2} - x^{3} y z^{2} - 17 x^{3} y z - 25 x^{3} y - 7 x^{3} z^{2} - 25 x^{3} z + 625 x^{3} + x^{2} y^{4} - 2 x^{2} y^{3} z + x^{2} y^{3} - 49 x^{2} y^{2} z - 245 x^{2} y^{2} - 2 x^{2} y z^{3} - 49 x^{2} y z^{2} - 610 x^{2} y z - 1875 x^{2} y + x^{2} z^{4} + x^{2} z^{3} - 245 x^{2} z^{2} - 1875 x^{2} z + 2 x y^{5} - x y^{4} z + 48 x y^{4} + x y^{3} z^{3} + 19 x y^{3} z^{2} + 66 x y^{3} z + 455 x y^{3} + 19 x y^{2} z^{3} + 336 x y^{2} z^{2} + 1245 x y^{2} z + 1875 x y^{2} - x y z^{4} + 66 x y z^{3} + 1245 x y z^{2} + 3750 x y z + 2 x z^{5} + 48 x z^{4} + 455 x z^{3} + 1875 x z^{2} + y^{6} - y^{5} z^{2} - 10 y^{5} z + 14 y^{5} - 23 y^{4} z^{2} - 237 y^{4} z - 19 y^{4} + 8 y^{3} z^{3} - 73 y^{3} z^{2} - 1564 y^{3} z - 625 y^{3} - y^{2} z^{5} - 23 y^{2} z^{4} - 73 y^{2} z^{3} + 366 y^{2} z^{2} - 1875 y^{2} z - 10 y z^{5} - 237 y z^{4} - 1564 y z^{3} - 1875 y z^{2} + z^{6} + 14 z^{5} - 19 z^{4} - 625 z^{3} \, . \end{align*}

Answer 2

If the Jacobian

$\frac{d(x_1, \cdots, x_n)}{d(t_1, \cdots, t_m) }$

is non-zero there is no relationship like:

$ P(x, y, z)=0 $.

In fact, the system

\begin{align*} x_1&=P_1(t_1, \cdots, t_m)\\ \vdots&\\ x_n&=P_n(t_1, \cdots, t_m) \end{align*}

is invertible;

we can consider the $t_1, \cdots, t_m $ as dependent variables and the $x_1, \cdots, x_n$ as independent.

We can give $x_1, \cdots, x_n$ any values and this shows that there is no relation like:

$ P(x, y, z)=0 $.