Asymptotic Differential Algebra and Model Theory of Transseries (eBook)

(AMS-195)

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven (Autoren)

eBook Download: PDF

2017
880 Seiten
Princeton University Press (Verlag)
978-1-4008-8541-1 (ISBN)

Infos

Matthias Aschenbrenner is professor of mathematics at the University of California, Los Angeles. Lou van den Dries is professor of mathematics at the University of Illinois, Urbana-Champaign. Joris van der Hoeven is director of research at the French National Center for Scientific Research (CNRS).

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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.

Produkt-Details

• Erscheint lt. Verlag: 6.6.2017
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 12 line illus.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Schulbuch / Wörterbuch ► Lexikon / Chroniken
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Schlagworte: abelian group • Absolute Galois Group • Additive map • Algebraically closed field • algebraic closure • algebraic differential equation • algebraic equation • algebraic extension • Algebraic Geometry • algebraic independence • Algebraic solution • Algebraic Structure • algebraic theory • algebraic variety • analytic function • angular component map • Annihilator (ring theory) • Asymptote • asymptotic analysis • asymptotic couple • asymptotic differential algebra • asymptotic equation • asymptotic expansion • asymptotic field • asymptotic relation • asymptotics • automorphism • axiomatic system • Axiom of choice • closed H-asymptotic couple • Closure (mathematics) • closure properties • coarsening • Cofinality • Commutative algebra • commutative property • Commutative Ring • Complexification (Lie group) • compositional conjugation • constant • Continuity • corollary • Decidability (logic) • Decomposition • definable set • Derivation • descriptive set theory • d-henselian • d-henselianity • Diagram (category theory) • Differential Algebra • differential equation • differential field • Differential Field Extension • differential Galois theory • differential-hensel • differential-henselian field • differential-henselianity • differentially closed field • Differential Module • Differential operator • differential polynomial • differential-valued extension • differential-valued field • directed union • dominant part • Embedding • Equalizer Theorem • Equation • Equivalence • eventual equalizer • eventual quantities • exceptional value • existential quantification • exponential function • exponential integral • exponentiation • extension • Factorization • Field Extension • Field of fractions • filtered algebra • filtered module • Foundations of mathematics • Function (mathematics) • gaussian extension • Graded algebra • grid-based transseries • Hahn Embedding Theorem • Hahn space • H-asymptotic couple • H-asymptotic field • henselian valued field • Hensel's lemma • H-field • homogeneous differential polynomial • Identity element • immediate extension • Inclusion map • Integral • integral domain • Integral element • integrally closed domain • Integration • Johnson's Theorem • Kähler differentials • Krull's Principal Ideal Theorem • linear differential equation • linear differential operator • linear differential polynomial • Liouville closed H-field • Liouville closure • Logarithmic derivative • low complexity • L-theory • Mathematical Induction • Mathematical problem • Mathematical Proof • mathematical structure • Mathematics • maximal immediate extension • model companion • model Theory • Module (mathematics) • Monotonic Function • monotonicity • Newton degree • Newton diagram • Newtonian differential field • newtonianity • newtonization • Newton-Liouville closure • Newton–Liouville closure • Newton multiplicity • Newton polynomial • Newton tree • Newton weight • Noetherian ring • ordered abelian group • ordered differential field • ordered set • ordered vector space • Order topology • Order type • polynomial • polynomial ring • Predicate (mathematical logic) • pre-differential-valued field • Principal ideal domain • Product topology • pseudocauchy sequence • pseudoconvergence • quantifier elimination • Quantifier (logic) • rational asymptotic integration • Recursive definition • regular local ring • residue field • Riccati transform • ring homomorphism • Scanlon's extension • Second-order Arithmetic • Semialgebraic set • Series (mathematics) • Set (mathematics) • simple differential ring • small derivation • Special Cut • Specialization • Subring • Subset • Substructure • Surjective function • System of polynomial equations • tensor product • Theorem • Topological space • Topology • transcendence degree • transfinite • Transfinite induction • Transitive relation • Transseries • triangular automorphism • triangular derivation • triangular linear map • Trichotomy (mathematics) • Tschirnhaus transformation • Union (set theory) • unitriangular group • unraveler • Valuation • Valuation ring • valuation topology • valued abelian group • valued differential field • Valued field • valued vector space • value group • Variable (mathematics) • Vector Space • Zariski topology • Zorn's lemma • λ-freeness • ω-freeness
• ISBN-10: 1-4008-8541-8 / 1400885418
• ISBN-13: 978-1-4008-8541-1 / 9781400885411