Executable Multivariate Polynomials


Title: Executable Multivariate Polynomials
Authors: Christian Sternagel, René Thiemann (rene /dot/ thiemann /at/ uibk /dot/ ac /dot/ at), Alexander Maletzky and Fabian Immler
Submission date: 2010-08-10
Abstract: We define multivariate polynomials over arbitrary (ordered) semirings in combination with (executable) operations like addition, multiplication, and substitution. We also define (weak) monotonicity of polynomials and comparison of polynomials where we provide standard estimations like absolute positiveness or the more recent approach of Neurauter, Zankl, and Middeldorp. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over polynomials. Our formalization was performed as part of the IsaFoR/CeTA-system which contains several termination techniques. The provided theories have been essential to formalize polynomial interpretations.

This formalization also contains an abstract representation as coefficient functions with finite support and a type of power-products. If this type is ordered by a linear (term) ordering, various additional notions, such as leading power-product, leading coefficient etc., are introduced as well.

Change history: [2010-09-17]: Moved theories on arbitrary (ordered) semirings to Abstract Rewriting.
[2016-10-28]: Added abstract representation of polynomials and authors Maletzky/Immler.
  author  = {Christian Sternagel and René Thiemann and Alexander Maletzky and Fabian Immler},
  title   = {Executable Multivariate Polynomials},
  journal = {Archive of Formal Proofs},
  month   = aug,
  year    = 2010,
  note    = {\url{http://isa-afp.org/entries/Polynomials.shtml},
            Formal proof development},
  ISSN    = {2150-914x},
License: GNU Lesser General Public License (LGPL)
Depends on: Abstract-Rewriting, Matrix
Used by: Groebner_Bases, Lambda_Free_KBOs