Formalization of the Lindemann-Weierstrass Theorem

Sophie Bernard (1, 2)

1 COMUE UCA - COMUE Université Côte d'Azur (2015-2019)
2 MARELLE - Mathematical, Reasoning and Software

This article details a formalization in Coq of the Lindemann-Weierstrass theorem which gives a transcendence criterion for complex numbers: this theorem establishes a link between the linear independence of a set of algebraic numbers and the algebraic independence of the exponentials of these numbers. As we follow Baker's proof, we discuss the difficulties of its formalization and explain how we resolved them in Coq. Most of these difficulties revolve around multivariate polynomials and their relationship with the conjugates of a univariate polynomial. Their study ultimately leads to alternative forms of the fundamental theorem of symmetric polynomials. This formalization uses mainly the Mathcomp library for the part relying on algebra, and the Coquelicot library and the Coq standard library of real numbers for the calculus part.

Mots clés: Coq; Formal proofs; Multivariate polynomials; Polynomial conjugates; Transcendance

Domaines: Informatique [cs]; Logique en informatique [cs.LO]; Logiciel mathématique [cs.MS]

Sophie Bernard. Formalization of the Lindemann-Weierstrass Theorem. Interactive Theorem Proving, Sep 2017, Brasilia, Brazil. ⟨hal-01647563⟩