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. Furthermore, a lot of generic properties of, and functions on, multivariate polynomials are formalized, including the substitution and evaluation homomorphisms, embeddings of polynomial rings into larger rings (i.e. with one additional indeterminate), homogenization and dehomogenization of polynomials, and the canonical isomorphism between R[X,Y] and R[X][Y].
[2010-09-17] Moved theories on arbitrary (ordered) semirings to Abstract Rewriting.
[2016-10-28] Added abstract representation of polynomials and authors Maletzky/Immler.
[2018-01-23] Added authors Haftmann, Lochbihler after incorporating their formalization of multivariate polynomials based on Polynomial mappings. Moved material from Bentkamp's entry "Deep Learning".
[2019-04-18] Added material about polynomials whose power-products are represented themselves by polynomial mappings.
Theories of Polynomials