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Asymptotic differential algebra and model theory of transseries. (English) Zbl 1430.12002
Annals of Mathematics Studies 195. Princeton, NJ: Princeton University Press (ISBN 978-0-691-17542-3/hbk; 978-0-691-17543-0/pbk; 978-1-4008-8541-1/ebook). xxi, 849 p. (2017).
This book has two complementary aims as indicated by the title:
to lay the foundations for the topic called by the authors “Asymptotic Differential Algebra”;
to give new important results about the model theory of the field of transseries \(\mathbb{T}\).
The field of transseries is informally introduced and contextualized in a very informative 26 pages “Introduction and Overview”, which recalls its role concerning Hilbert’s 16th problem about differential equations. It has also been introduced in model theory as a nonarchimedean model of the o-minimal theory of the field of real numbers with exponential function. \(\mathbb{T}\) is precisely defined (in particular, the version that was before called Log-Exp series field) in a first Appendix. Roughly speaking, \(\mathbb{T}\) is an ordered exponential field consisting of power series, obtained by closing the ordered field of rational functions \(\mathbb{R}(x)\) (with \(x>\mathbb{R}\)) by taking real powers, exp, log and infinite sums with well-ordered supports. In particular, any transseries is exponentially bounded, with finite exponential height and logarithmic depth. This ordered field is endowed with its natural valuation (the valuation ring is the convex hull of \(\mathbb{R}\)), a natural term by term derivation, and also with various other operations, e.g. composition by real analytic functions for “bounded” transseries.
Asymptotic differential algebra is developed in the first 10 chapters in a self-contained way. In Chapters 1 to 5, the authors gather classical and/or needed notions and results from algebra, in particular commutative and differential algebra: ordered and valued fields (equicharacteristic 0), abelian ordered and valued groups, differential polynomials. A particular emphasis is put on henselian valued fields and spherically complete valued fields, especially generalized series fields (in the sense of H. Hahn, that is formal power series with well-ordered support). They are of crucial importance of course in the construction of \(\mathbb{T}\), but also more generally as universal domains for valued fields, and for explicit algebraic resolution of equations. Note that in most of the book, valuations are replaced equivalently by dominance relations \(\preccurlyeq\) in the sense of G. H. Hardy. Chapters 4 and 5 on differential algebra deal with decomposition of differential operators or polynomials “à la Ritt-Kolchin”, and its links with differential rings and ideals, and differential fields extensions. The setting their and for the rest of the book is fields of characteristic 0 with one distinguished derivation. Also, the notions of additive and multiplicative conjugations (\(Y\mapsto f+Y\) and \(Y\mapsto fY\), corresponding to generalized blowups possibly with translation) are introduced: they subsequently play a prominent role in the differential henselian context.
Chapters 6 to 10 deal with the central objects of asymptotic differential algebra, namely valued differential fields (Chapters 6 to 9) and in particular ordered valued differential fields (Chapter 9 and 10). The authors gather several results: extensions (immediate, algebraic, transcendental, etc.) of such fields, differential-henselian fields (i.e. fields for which a differential analogue of the implicit function theorem holds). Note that the derivation is assumed to be small (i.e. to preserve the valuation ideal: this can be obtained harmlessly by a change of derivation via a multiplicative conjugation). Then, following M. Rosenlicht’s extensive work on Hardy fields, they introduce and study:
asymptotic fields, i.e. valued differential fields for which a strong version of l’Hospital rule holds: \[ \forall f,g,\, v(f),v(g)\neq 0,\ v(f)>v(g)\Leftrightarrow v(f')>v(g') ; \]
\(H\)-fields (extensively studied by the first two authors over the past 20 years), i.e. ordered valued differential fields that are an axiomatic version of Hardy fields:by definition, the valuation ring is exactly the convex hull of the subfield of constants, and infinitely increasing elements have positive derivatives. Hence, \(H\)-fields are asymptotic fields with also a rule for the logarithmic derivatives \[ \forall f,g,\ v(f)>v(g)>0 \Rightarrow v(f'/f)\leq v(g'/g). \] As for Hardy fields, they admit a Liouville closure, that is an extension which is closed under resolution of first order linear differential equations. But this extension might not be unique.
Key properties between the derivation and the valuation are also encoded at the level of the value group via the notion of asymptotic couple (notion borrowed from M. Rosenlicht and also extensively studied by the first two authors).
The remaining Chapters 11 to 16 are mostly devoted to gathering and proving recent or new important results about the model theory of H-fields and the field of transseries \(\mathbb{T}\). A primer on (basic 1st order) model theory is provided in the second appendix. The first key notion is the one of \(\omega\)-freeness, which encodes in first order terms the possibility to solve certain second order linear differential equations with nonoscillating solutions (extending ideas already developed by M. Rosenlicht and M. Boshernitzan for Hardy fields). It is a refinement of the notion of \(\lambda\)-freeness previously studied by the authors, and that deals with logarithmic integration. Contrary to the latter, the \(\omega\)-freeness property guarantees the uniqueness of the Liouville closure. It also has nice computational consequences concerning the following second key notion. Based on the explicit methods for solving equations “à la Newton-Puiseux” with transseries that he developed for 20 years, the 3rd author has introduced the notion of newtonianity. It axiomatizes the possibility of solving henselian differential equations under asymptotic conditions. Three other functions are introduced: \(\iota\) which renders the multiplicative inverse, and \(\Lambda\) and \(\Omega\) which are linked to resolution of 1st and 2nd order linear differential equations. The main results of the authors can be summarized as:
Theorem. The first order theory of \(\mathbb{T}\) as an ordered valued differential field is exactly the one of Liouville closed \(H\)-fields with small derivation that are \(\omega\)-free and Newtonian. This theory is model complete. Moreover, if one adds the functions \(\iota\), \(\Lambda\) and \(\Omega\) to the language, one has elimination of quantifiers.
The authors derive geometric and algebraic tameness consequences:
any model of this theory is existentially closed (i.e. it has no proper algebraic differential extension);
the theory of \(\mathbb{T}\) is o-minimal at infinity (i.e. any definable set with arbitrary large values actually possesses a final segment);
the definable subsets of \(\mathbb{R}^n\) are the semi-algebraic sets;
the theory has the Non Independence Property (NIP: a combinatorial notion of tameness due to S. Shelah).
More recently, the authors have showed that being \(\omega\)-free and newtonian is equivalent for Liouville closed \(H\)-fields to having the Differential Intermediate Value Property: if a differential polynomial changes sign, then it has an actual solution. This completes the parallel that one can make between \(\mathbb{R}\) for real algebra and geometry, and \(\mathbb{T}\) for real differential algebra and geometry: the authors have reached a milestone toward the conjecture that \(\mathbb{T}\) is a universal domain for Hardy fields, and more generally for asymptotic differential algebra.

MSC:
12-02 Research exposition (monographs, survey articles) pertaining to field theory
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
12L12 Model theory of fields
12H05 Differential algebra
03C64 Model theory of ordered structures; o-minimality
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