On Sylvester's problem and Haar spaces (Q791058)

Given a finite set of points in the plane (with distinct x coordinates) must there exist a polynomial of degree n that passes through exactly \(n+1\) of the points? Provided that the points do not all lie on the graph of a polynomial of degree n then the answer to this question is yes. This generalization of Sylvester's problem (the \(n=1\) case) is established as a corollary to a version of Sylvester's Problem that holds for certain finite dimensional Haar spaces of continuous functions.