Possibility for two distinct polynomials to interpolate the same m data points

Question

Is it ever possible for two distinct polynomials to interpolate the same m data points? If so, under what conditions, and if not, why?

Answer 1

Yes, it is possible. The points $(-1, 0)$ and $(1,0)$ are interpolated by each of the two polynomials: $$ p(x) = x^2 - 1 $$ and $-p(x)$.

There are much more interesting examples.

One of the general reasons as to why this should be possible: there is a problem of Lagrangian or Newtonian interpolation (we seek a polynomial going through the given points), but there is also the problem of Hermitian interpolation which, in addition to the requirement of going through the given points, also requires the polynomial to have prescribed derivatives at these points.

So, a polynomial that solves an Hermitian interpolation problem for a given set of points also solves the Lagrangian interpolation problem for those points. But not vice versa.