By Matthias Aschenbrenner,Lou van den Dries,Joris van der Hoeven
Asymptotic differential algebra seeks to appreciate the suggestions of differential equations and their asymptotics from an algebraic viewpoint. The differential box of transseries performs a valuable function within the topic. along with powers of the variable, those sequence may possibly include exponential and logarithmic phrases. over the past thirty years, transseries emerged variously as super-exact asymptotic expansions of go back maps of analytic vector fields, in reference to Tarski's challenge at the box of reals with exponentiation, and in mathematical physics. Their formal nature additionally makes them appropriate for computing device computations in desktop algebra systems.
This self-contained publication validates the instinct that the differential box of transseries is a common area for asymptotic differential algebra. It does so by means of developing within the realm of transseries an entire removal idea for structures of algebraic differential equations with asymptotic aspect stipulations. starting with historical past chapters on valuations and differential algebra, the booklet is going directly to boost the fundamental conception of valued differential fields, together with a idea of differential-henselianity. subsequent, H-fields are singled out between ordered valued differential fields to supply an algebraic environment for the typical homes of Hardy fields and the differential box of transseries. The examine in their extensions culminates in an analogue of the algebraic closure of a box: the Newton-Liouville closure of an H-field. This paves how you can a quantifier removal with attention-grabbing consequences.
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