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 $0$.

When you have a redundant system of $N$ equations of any type for $n$ variables such$$p_i=p_i(x_1,\dots,x_n)=0\qquad i=1,2,\cdots N >n$$ one of the methods for solving consists in the minimization of $$\Phi(x_1,\dots,x_n)=\sum_{i=1}^N p_i^2$$ So, if the result is $0$, then the system is redundant and $\Phi$ contains all the roots.
Now, consider the case of polynomials and see what is $\Phi$. 
