A Generalization

I explore here a slight generalization to this theorem.
Given positive integers k, m and n with n < k, and k polynomials p_i each in n variables, each of degree ≤ m; there exists a non-zero polynomial q in k variables of degree ≤ m such that for any combination of n reals, λ₀, … λ_n−1
q(p₀(λ₀, … λ_n−1), … p_k−1(λ₀, … λ_n−1)) = 0.
We seek a proof along the same lines.

From the perspective of algebraic varieties this says that a parametrically defined n dimensional manifold in a k dimensional space, defined by n parameters λ₀, … λ_n−1, is a variety whose degree is no more than the maximum of the degrees of the defining functions.

For the general polynomial q in k variables of degree ≤ m the expression above would be a polynomial in n variables of degree ≤ m². The challenge is to find such a q of degree ≤ m. For 0≤i<k we denote p_i(λ₀, … λ_n−1) by x_i. Thus we seek non-zero q with degree ≤ m such that q(x₀, … x_k−1) = 0.

This code seems to refute the conjecture.

To find our q that makes q(x₀, … x_k−1) vanish we consider kn expressions for each combination of i and j where 0≤i<k and 0≤j<n
λ_jⁱx_i − λ_jⁱp_i(λ₀, … λ_n−1)
Each of these are 0 by definition of x_i. We have thus a set of kn equations that are linear in

In these expressions the k variables x_i become the formal parameters of the polynomial q that we seek. In this role we have for 0≤i<k
x_i = p_i(λ₀, … λ_n−1)

Let q_{i₀, … i_k−1} be one of the coefficients of the polynomial q:
q(x₀, … x_k−1) = Σ q_{i₀, … i_k−1} x₀^i₀x₁^i₁…x_k−1^i_k−1 where the sum is over combinations of non-negative integer values of i_j with the sum of the i’s ≤ m.