We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. We then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

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AppendicesAppendix A: Computing tropical solutions to the tropicalization of some linear p-adic ODEs Résumé de thèse en français vi "Theorems are like sausages, it is better not to see them being made" -(almost) Otto von Bismarck

An algebraic system consisting of a set together with an associative binary and a ternary heap operations is studied. Such a system is termed a pre-truss and if a binary operation distributes over the heap operation on one side one speaks about a near-truss. If the binary operation in a near-truss is a group operation, then it can be specified or retracted to a skew brace, the notion introduced in [L. Guarnieri & L. Vendramin, Math. Comp. 86 (2017), 2519-2534. On the other hand if the binary operation in a near-truss has identity, then it gives rise to a skew-ring as introduced in [W. Rump, J. Algebra Appl. 18 (2019), 1950145]. Congruences in pre-and near-trusses are shown to arise from normal sub-heaps with an additional closure property of equivalence classes that involves both the ternary and binary operations. Such sub-heaps are called paragons. A necessary and sufficient criterion on paragons under which the quotient of a unital near-truss corresponds to a skew brace is derived. Regular elements in a pre-truss are defined as elements with left and right cancellation properties; following the ring-theoretic terminology pre-trusses in which all non-absorbing elements are regular are termed domains. The latter are described as quotients by completely prime paragons, also defined hereby. Regular pre-trusses and near-trusses as domains that satisfy the Ore condition are introduced and the pre-trusses of fractions are constructed through localisation. In particular, it is shown that near-trusses of fractions without an absorber correspond to skew braces.