Question

Prove that n distinct point-value pairs are necessary to uniquely specify a polynomial of degree-bound n, that is, if fewer than n distinct point-value pairs are given, they fail to specify a unique polynomial of degree-bound n.

Solution

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Let we have n1n-1 distinct points, hence we can get a unique polynomial representation for a polynomial up to degree (n1n-1), satisfied by the n1n-1 points, according to the mentioned theorem in the hint.

Now if we add a new point to our (n1n-1) points we can get a new unique representation for a polynomial up to degree nn, satisfied by thhe nn points.

Again if we add a new point (different from the point just we chose) to our (n1n-1) points we can get a new unique representation for a polynomial up to degree nn, satisfied by thhe nn points.

Since our n1n-1 points give two different representation of two different polynomial, n1n-1 points can't give a unique representation for a polynomial of degree nn.

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