Does there exist a finite set of homogeneous polynomials (+ property) whose unique solution is equivalent to a finite sequence of naturals?

Question

Consider the set $\{2,3,5\}$ of natural numbers. Letting $p = 2, q = 3, r = 5$ we have: the polynomial equations: $$p + q = r, \\ p^2 + r = q^2 \\ q^3 - p = r^2 $$

Each is a homogeneous polynomials such that each variable occurs only once in its expanded expression. Is there a finite set of such polynomial equations such that the only solution is $(p,q,r) = (2,3,5)$?

Answer 1

Your last two equations involve non-homogeneous polynomials. And if I understand your question, the answer is no. For if $(a,b,c)$ is a solution of an equation $P(x,y,z)=0$ where $P$ is a homogeneous polynomial, then $(ka,kb,kc)$ is also a solution for any constant $k$.