### Abstract

A well known result from algebra is that, on any field, there is exactly one polynomial of degree less than n interpolating n points [1, §7].

This entry contains a formalization of the
above result, as well as the following generalization in the case of
finite fields *F*: There are
*|F| ^{m-n}* polynomials of degree
less than

*m ≥ n*interpolating the same n points, where

*|F|*denotes the size of the domain of the field. To establish the result the entry also includes a formalization of Lagrange interpolation, which might be of independent interest.

The formalized results are defined on the algebraic structures from HOL-Algebra, which are distinct from the type-class based structures defined in HOL. Note that there is an existing formalization for polynomial interpolation and, in particular, Lagrange interpolation by Thiemann and Yamada [2] on the type-class based structures in HOL.