How to verify that one of equations in a polynomial system is redundant? I know that system of polynomial equations
$$
p_1(x_1,\dots,x_n)=0,..., p_N(x_1,\dots,x_n)=0
$$
has infinitely many solutions.
I computed  some of them numerically and notices that they always satisfy one more polynomial equation 
$$
q(x_1,\dots,x_n)=0.
$$
I would like to prove that this is always the case. 
Question:  Does it exist a method to prove that
that $q(x_1,\dots,x_n)=0$ follows from $p_1(x_1,\dots,x_n)=0,..., p_N(x_1,\dots,x_n)=0$?
Extra info:
in my application $p_1, p_N$ contains terms of degrees $3$ and $0$, and $q$  contains terms of degree $6$,$3$, and $0$.

Since the question is not very concrete I will give a broad answer: Given your polynomial equations $p_1,\ldots,p_N$ you can associate to them an ideal 
$$ I:=\left\{\sum_{i=1}^N q_ip_i\vert q_i \in k[x_1,\ldots,x_n]   \right\} $$
In plain english: $I$ contains every linear combination of the $p_i$ with coefficients that are polynomials too. Now you can easily see that every element of $I$ vanishes on the zero locus of the $p_i$. We say that the $p_i$ generate $I$. 
Now your question can be interpreted in 2 ways: you can either ask if there is a algorithm to decide if the $p_i$ are a minimal generating system of $I$. The answer to this is that it's known that in general there is no such algorithm. See for example here. However there my be cases in which you can decide this. But we can say nothing about this until you give us more information.
The second interpretation of your question would be to ask given a polynomial $q$, how to decide if $q^m \in I$ for some $m \in \mathbb{N}$? Again I don't know about a general answer to this question. 
