Suppose that $P_k(x,y)$ is a symmetric polynomial with real coefficients (let's say for now that it is of degree 4, but I am curious about results for general degrees). In other words, suppose that $$P_k(x,y) = a_kx^2y^2 + b_k(x^2y + xy^2) + c_k(x^2+y^2) +d_kxy + e_k(x+y) + f_k.$$ Now, suppose I have a system of equations $$P_i(x,y) = P_i(x', y')$$ for $i = \{1, 2, \dots, m\}.$ In the setting I care about, I have that $x,y,x', y' \in [0,1],$ though probably this condition isn't too important.
Could you give some general conditions on the $P_i$ and $m$ that would then force $x=x', y=y'$? I imagine you might need that the polynomials are sort of "linearly independent," but I'm not quite sure how to make this rigorous. Thanks!