I explore here a slight generalization to this theorem.
Given positive integers k, m and n with n < k, and k polynomials pi 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, … λn−1
q(p00, … λn−1), … pk−10, … λn−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, … λn−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 pi0, … λn−1) by xi. Thus we seek non-zero q with degree ≤ m such that q(x0, … xk−1) = 0.

This code seems to refute the conjecture.

To find our q that makes q(x0, … xk−1) vanish we consider kn expressions for each combination of i and j where 0≤i<k and 0≤j<n
λjixi − λjipi0, … λn−1)
Each of these are 0 by definition of xi. We have thus a set of kn equations that are linear in


In these expressions the k variables xi become the formal parameters of the polynomial q that we seek. In this role we have for 0≤i<k
xi = pi0, … λn−1)
Let qi0, … ik−1 be one of the coefficients of the polynomial q:
q(x0, … xk−1) = Σ qi0, … ik−1 x0i0x1i1…xk−1ik−1 where the sum is over combinations of non-negative integer values of ij with the sum of the i’s ≤ m.