Deciding irrationality and transcendence – DeRerumNatura

Classifying the objects of a mathematical theory requires to make its predicates effective and to automate its computations efficiently so that they are feasible on concrete instances. This is what we propose to do in order to solve problems in relation to numbers, analytic functions, and generating series. We want indeed to make effective, automatic, and efficient the classification of certain objects in the framework of field-proven in number theory (E-functions, G-functions, Mahler functions) and in combinatorics (constrained walks). We tackle these topics in a single project as the study of the underlying functional equations will benefit from the same algorithmic tools (Galois theory, integration, Gröbner bases, explicit formula reconstruction). Beside striking results on numbers and walks, we expect that the general scope of the tools developed will have a broad and lasting impact.