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.
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