### 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.

### License

BSD License

### Topics

- Mathematics/Algebra

### Session Interpolation_Polynomials_HOL_Algebra

- Bounded_Degree_Polynomials
- Lagrange_Interpolation
- Interpolation_Polynomial_Cardinalities

### Used by

- Formalization of Randomized Approximation Algorithms for Frequency Moments
- Universal Hash Families

## Cite

×

Interpolation_Polynomials_HOL_Algebra-AFP

    @article{Interpolation_Polynomials_HOL_Algebra-AFP,
      author  = {Emin Karayel},
      title   = {Interpolation Polynomials (in HOL-Algebra)},
      journal = {Archive of Formal Proofs},
      month   = {January},
      year    = {2022},
      note    = {\url{https://isa-afp.org/entries/Interpolation_Polynomials_HOL_Algebra.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

Download

## Download

× Download latest

Older releases:

- Sep 13, 2023 : Isabelle2023
- Oct 27, 2022 : Isabelle2022
- Jan 31, 2022 : Isabelle2021-1