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My question concerns a multivariate generalization of this. Specifically, for integer $n\geq 2$, I have parametric polynomial equations of the form

$$x_{1} = p_{1}(t_1, t_2, ..., t_{n-1}),\\ x_{2} = p_{2}(t_1, t_2, ..., t_{n-1}),\\ \vdots\\ x_{n} = p_{n}(t_1, t_2, ..., t_{n-1}),$$ where $p_{1}, p_{2}, ..., p_{n}$ are all homogeneous polynomials in $(n-1)$ parameter variables $t_{1}, ..., t_{n-1}$ with coefficients $\pm 1$.

Let $q(x_1, x_2, ..., x_n)=0$ be the implicit equation obtained after the elimination of the parameters $t_{1}, ..., t_{n-1}$. Is it true that $q$ is a polynomial in $x_1, x_2, ..., x_n$? If true, is it possible to say anything about the degree of the polynomial $q$ in terms of the degrees of $p_{1}, ..., p_{n}$? In case it helps, in my case, $p_1, ..., p_n$ have degrees $1,2,...,n$, respectively.

Any pointer/reference will be much appreciated.

Abhishek Halder
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  • Have you looked at the reference of Cox, Little, and O'Shea given in the linked question? I cited one of the theorems in that book here. – Viktor Vaughn Dec 07 '20 at 05:13
  • @ Richard D. James: My area is far from algebraic geometry. Looking at the reference, am I correct to understand that $q$ is guaranteed to be a polynomial but explicitly computing it as a multidimensional resultant is perhaps difficult? – Abhishek Halder Dec 08 '20 at 09:19

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