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Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems.This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Series, Arithmetic. --- Divergent series. --- Asymptotic expansions. --- Differential algebra. --- Algebra, Differential --- Differential fields --- Algebraic fields --- Differential equations --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Series, Divergent --- Series --- Arithmetic series --- Progressions, Arithmetic --- Equalizer Theorem. --- H-asymptotic couple. --- H-asymptotic field. --- H-field. --- Hahn Embedding Theorem. --- Hahn space. --- Johnson's Theorem. --- Krull's Principal Ideal Theorem. --- Kähler differentials. --- Liouville closed H-field. --- Liouville closure. --- Newton degree. --- Newton diagram. --- Newton multiplicity. --- Newton tree. --- Newton weight. --- Newton-Liouville closure. --- Riccati transform. --- Scanlon's extension. --- Zariski topology. --- algebraic differential equation. --- algebraic extension. --- angular component map. --- asymptotic couple. --- asymptotic differential algebra. --- asymptotic field. --- asymptotic relation. --- asymptotics. --- closed H-asymptotic couple. --- closure properties. --- coarsening. --- commutative algebra. --- commutative ring. --- compositional conjugation. --- constant. --- continuity. --- d-henselian. --- d-henselianity. --- decomposition. --- derivation. --- differential field extension. --- differential field. --- differential module. --- differential polynomial. --- differential-hensel. --- differential-henselian field. --- differential-henselianity. --- differential-valued extension. --- differentially closed field. --- dominant part. --- equivalence. --- eventual quantities. --- exponential integral. --- extension. --- filtered module. --- gaussian extension. --- grid-based transseries. --- henselian valued field. --- homogeneous differential polynomial. --- immediate extension. --- integral. --- integrally closed domain. --- linear differential equation. --- linear differential operator. --- linear differential polynomial. --- mathematics. --- maximal immediate extension. --- model companion. --- monotonicity. --- noetherian ring. --- ordered abelian group. --- ordered differential field. --- ordered set. --- pre-differential-valued field. --- pseudocauchy sequence. --- pseudoconvergence. --- quantifier elimination. --- rational asymptotic integration. --- regular local ring. --- residue field. --- simple differential ring. --- small derivation. --- special cut. --- specialization. --- substructure. --- transseries. --- triangular automorphism. --- triangular derivation. --- valuation topology. --- valuation. --- value group. --- valued abelian group. --- valued differential field. --- valued field. --- valued vector space.