Leseprobe
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Asymptotic Differential Algebra and Model Theory of Transseries
Lou van den Dries, Matthias Aschenbrenner, Joris van der Hoeven
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Differential algebra, Function (mathematics), Principal ideal domain, Quantifier (logic), Algebraic extension, Closure (mathematics), Module (mathematics), Axiom of choice, Existential quantification, Transfinite induction, Algebraic closure, Identity element, Model theory, Complexification (Lie group), Mathematical structure, Tensor product, Set (mathematics), Substructure, Logarithmic derivative, Asymptotic expansion, Automorphism, Analytic function, Diagram (category theory), Differential equation, Subring, Residue field, Equation, Trichotomy (mathematics), Ring homomorphism, Ordered vector space, Field extension, Exponentiation, Order topology, Differential operator, Vector space, Theorem, Newton polynomial, L-theory, Subset, Quantifier elimination, Topology, Predicate (mathematical logic), Descriptive set theory, Asymptote, Order type, Transcendence degree, Algebraic equation, Asymptotic analysis, Embedding, Mathematical induction, Algebraic theory, Zorn's lemma, Surjective function, Commutative ring, Commutative algebra, Product topology, Abelian group, System of polynomial equations, Valuation ring, Polynomial, Annihilator (ring theory), Linear differential equation, Variable (mathematics), Algebraic differential equation, Algebraic independence, Corollary, Differential Galois theory, Integral domain, Hahn embedding theorem, Commutative property