The Mason–Stother's Theorem

 

Title: The Mason–Stother's Theorem
Author: Manuel Eberl
Submission date: 2017-12-21
Abstract:

This article provides a formalisation of Snyder’s simple and elegant proof of the Mason–Stothers theorem, which is the polynomial analogue of the famous abc Conjecture for integers. Remarkably, Snyder found this very elegant proof when he was still a high-school student.

In short, the statement of the theorem is that three non-zero coprime polynomials A, B, C over a field which sum to 0 and do not all have vanishing derivatives fulfil max{deg(A), deg(B), deg(C)} < deg(rad(ABC)) where the rad(P) denotes the radical of P, i. e. the product of all unique irreducible factors of P.

This theorem also implies a kind of polynomial analogue of Fermat’s Last Theorem for polynomials: except for trivial cases, An + Bn + Cn = 0 implies n ≤ 2 for coprime polynomials A, B, C over a field.

BibTeX:
@article{Mason_Stothers-AFP,
  author  = {Manuel Eberl},
  title   = {The Mason–Stother's Theorem},
  journal = {Archive of Formal Proofs},
  month   = dec,
  year    = 2017,
  note    = {\url{http://isa-afp.org/entries/Mason_Stothers.html},
            Formal proof development},
  ISSN    = {2150-914x},
}
License: BSD License
Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.