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

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    Assume polynomial $p(x)$ interpolate a list of $m$ data points $(x_1,y_1),\ldots,(x_m,y_m)$. Another polynomial $q(x)$ will interpolate same list of data points when and only when their difference $p(x) - q(x)$ is divisible by the polynomial $\prod_{k=1}^m (x - x_k)$. This means there is at most one polynomial which interpolate the data points with degree less than $m$. – achille hui Apr 25 '19 at 20:25

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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.

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