Abstract

We formalize algebraic numbers in Isabelle/HOL, based on existing libraries
for matrices and Sturm’ s theorem. Our development serves as a verified
implementation for real and complex numbers, and it admits to compute roots
and completely factor real and complex polynomials, provided that all
coefficients are rational numbers. Moreover, we provide two implementations
to display algebraic numbers, an injective and expensive one, and a faster
but approximative version.
To this end, we mechanize several results on resultants, which also required
us to prove that polynomials over a unique factorization domain form again
a unique factorization domain. We moreover formalize algorithms for
factorization of integer polynomials: Newton interpolation, factorization
over the integers, and Kronecker’ s factorization algorithm, as well as a
factorization oracle via Berlekamp’ s algorithm with the Hensel lifting.

BibTeX

@inproceedings{RTAY2016-ITP,
  author    = "Ren{\'e} Thiemann and Akihisa Yamada",
  title     = "Algebraic Numbers in Isabelle/HOL",
  booktitle = "Proceedings of the 7th International Conference on
               Interactive Theorem Proving",
  editor    = "Jasmin Christian Blanchette and Stephan Merz",
  series    = "Lecture Notes in Computer Science",
  volume    = 9807,
  pages     = "391--408",
  year      = 2016,
  doi="10.1007/978-3-319-43144-4_24"
}