An algebraic curve of degree n here is defined as the points in a plane satisfying p(x, y) = 0 where p is a non-zero polynomial of degree n in two variables.

The general conjecture is:
If n > 0 and xp and yp are both polynomials of degree ≤ n
then the plane curve defined parametrically by p(t) = (xp(t), yp(t)) is an algebraic curve of degree ≤ n.

A purely algebraic statement of the conjecture is that if p and q are polynomials of degree ≤ n then there is a non-zero polynomial r in two variables of degree ≤ n such that for all t, r(p(t), q(t)) = 0.

(This seems to be proven here.)
I struggle over a generalization here and a proof that I understand better.