Formal Proofs of Transcendence for e and $\pi$ as an Application of Multivariate and Symmetric Polynomials

We describe the formalisation in Coq of a proof that the numbers e and $\pi$ are transcendental. This proof lies at the interface of two domains of mathematics that are often considered separately: calculus (real and elementary complex analysis) and algebra. For the work on calculus, we rely on the Coquelicot library and for the work on algebra, we rely on the Mathematical Components library. Moreover, some of the elements of our formalized proof originate in the more ancient library for real numbers included in the Coq distribution. The case of $\pi$ relies extensively on properties of multivariate polynomials and this experiment was also an occasion to put to test a newly developed library for these multivariate polynomials.