This is a cross-post from MSE where it did not receive a response.

For integer $n\geq 2$, consider a parameterization of the coordinates $(x_1, x_2, ..., x_{n})$ in terms of the parameters $(s_{1},s_{2}, ..., s_{n-1})$ given by $$x_{j} = \displaystyle\sum_{i=1}^{n-1}(-1)^{i+1}s_{i}^{n-j+1}, \qquad j=1, ..., n.$$ We want to find the implicitization $p(x_1, x_2, ..., x_{n}) = 0$ by eliminating $s_{1},s_{2}, ..., s_{n-1}$.

For example, when $n=2$, the parameterization becomes $$x_{1} = s_{1}^{2},\\ x_{2} = s_{1},$$ and the implicit equation is $p(x_1,x_2) \equiv x_{2}^{2}-x_{1} = 0$.

When $n=3$, the parameterization becomes $$x_{1} = s_{1}^{3} - s_{2}^{3},\\ x_{2} = s_{1}^{2} - s_{2}^{2},\\ x_{3} = s_{1} - s_{2},$$ and elementary algebra gives the implicit equation $p(x_1,x_2,x_3) \equiv x_{3}^{4}-4x_{3}x_{1}+3x_{2}^{2} = 0$.

When $n=4$, the parameterization becomes $$x_{1} = s_{1}^{4} - s_{2}^{4} + s_{3}^{4},\\ x_{2} = s_{1}^{3} - s_{2}^{3} + s_{3}^{3},\\ x_{3} = s_{1}^{2} - s_{2}^{2} + s_{3}^{2},\\ x_{4} = s_{1} - s_{2} + s_{3}.$$

**Question:** How to systematically derive a general formula for $p(x_1,x_2,...,x_n)$?

**Two comments:**

(i) I have tried looking into the Macaulay resultant but it is not clear to me if this helps to get a general formula in terms of $n$.

(ii) If we had all plus signs in the $s$ monomials, then the parameterization would reduce to power sums, and we could use Newton's identities to get the implicitization. But I am not sure if that can be generalized here.