
These are notes from Lou van den Dries' lectures on the model theory of Hardy fields, during the Fields thematic semester on tame geometry, JanuaryJune 2022.
The notes are incomplete, and start with considerations on second order linear equations over Hardy fields. In particular, the material on Hausdorff and Hardy fields and semialgebraic differential equations of order 1 over Hardy fields is not here. So until I include it (which I may never do), the notes here start from page 13.
Abstract 5
1Hausdorff fields 9
1.1 9
1.2 9
1.3 9
2Hardy fields 11
2.1 11
2.2 11
3Linear equations of order 13
Fact 14
What do we know about ? 14
4Cuts and differential algebra 17
4.1Asymptotic couples 17
4.1.1Asymptotic couples 17
Basic facts 17
4.1.2Further basic facts about couples 18
Trichotomy for asymptotic couples 19
4.3Cuts and Hfields 20
4.3.1Hfields 20
4.3.2Cuts in Hfields 21
4.3.3The function 21
4.4Differential algebra 22
4.4.1Differential polynomials 22
4.4.2Linear differential operators 22
4.4.3The case of differential fields 22
4.4.4Compositional conjugation 23
4.5Eventual behavior 23
4.5.1Newton degree of a differential polynomial 24
5Hclosed Hardy fields 25
5.1Back to Hardy fields 25
5.1.1Some preleminary observations: 25
5.1.2Trailing linear differential operators 25
5.1.3Sketch of proof of the main theorem 26
5.2Exponential sums over Hardy fields 28
5.2.1The universal exponential extension 28
5.2.2Exponential sums and linear differential operators 29
6Filling gaps in Hardy fields 31
6.1Countable pseudoCauchy sequences 31
6.2Pseudolimits in Hausdorff fields 32
6.3[A few missing notes] 32
6.4Filling cuts in the value group 33
Bibliography 35
Index 37
Glossary 39
0217: Lecture 10
Notation
Proposition
(3.0.1) 
and is nonoscilllating for all .
So generates a Hardy field over .
Lemma
Proof. We have since . Let with and for all . It is enough to show the result for the solution
of (3.0.1). Set for all . So for , we have
where for , we have
It follows that for . In particular . Routine computations show that between and , we also have , so as a germ.
Lemma
Proof. Changing with and with if necessary, we can arrange that . For , set . We have for some polynomial with integer coefficients. We still have , whence since is Hardian. In particular satisfies the conditions of Lemma 3.0.3, whence . In particular .
Theorem
for some and , or
for all and .
In case I, there is a unique with which generates a Hardy field over . In case II, every with generates a Hardy field over .
Proof. Suppose that we are in case , with corresponding . Setting , the ODE
transforms into
Then Proposition 3.0.2 gives the result. Suppose now that we are in case II. Let with . We will show that is nonoscillating for all . For , we have
so Lemma 3.0.4 gives that, whence in particular is nonoscillating. Therefore generates a Hardy field over .
Remark
Suppose we are in case II of the theorem. Then there are continuummany solutions witnessing case II, any two of which are incompatible (because of the oscillating nature of nonzero solutions of the homogeneous harmonic equation). Plausible: in case II, for any two solutions , the fields and are isomorphic.
Does case II actually occur? Boshernitzan show that this is the case in for . One line in Boshernitzan's proof should be made clearer. Indeed Boshernitzan uses the following fact about complex linear differential equations:
which is holomorphic on a nonempty open extends holomorphicaly to . So come back to
we note that is entire, so any solution extends into an entire function. This closes the gap in Boshernitzan's proof (see upcoming notes from Lou).
Let be a Hardy field. A germ is said Hardian over , or Hardian if lies in some Hardy field extension of . Boshernitzan defines a larger Hardy field
Equivalently, this is the intersection of all maximal Hardy fields containing .
Let us focus on , which is the set of “most Hardian germs”, i.e. germs that are contained in all maximal Hardy fields. By Boshernitzan's result, the ODE
has no solution in . The differential field is differentially algebraic (over say). We'll give a sketch of proof shortly. As a consequence, no is transexponential or sublogarithmic. Moreover is closed under composition [3, Theorem 6.8].
Fernando Sanz suggests looking at the differential equation
whose solutions are supposed to be definable in minimal structures.
Question
Answer
Question
Question
Sketch of proof that is differentially algebraic. Suppose is Hardian but not differentially algebraic. One (Boshernitzan, for instance) can show that for any sufficiently small i.e. if for all , then is also Hardian. But then there is a maximal Hardy field containing , which does not contain : a contradiction.
General fact. [in a paper of Lou and Matthias]Let be a Hardy field containing . Let . Then there is an with either
for all Hardian germs ,or
for all .
0222: Lecture 11
Let be a Hardy field, let be the natural valuation of seen as an ordered field. We write , or sometimes , for the corresponding valuation ring. Recall that for nonzero , we have . So we have an operation on the value group
We also have a function
which will have useful properties. This function and the structure were introduced by M. Rosenlicht. The dagger operation is in particular a valuation on the ordered group , that is, for with , we have . The properties of on showed earlier also imply that for and , we have and .
Definition
If , then .
for all .
If , then .
If in addition, we have , then we call and asymptotic couple. We will often write and for all . We say that has small derivation if for all .
So the value group of with the dagger operation defined above is an asymptotic couple. In fact the same holds if is any Hfield with small derivation. It is sometimes convenient to extend to a function by setting and . This preserves the axioms above.
The basic facts about asymptotic couples and asymptotic couples were either derived by Rosenlicht or proved in [2, Sections 6.5, 9.1 and 9.2].
If and , then , i.e. .
The function is strictly increasing.
If , then .
A consequence of is that extends uniquely to the divisible hull of in such a way that the corresponding structure is an asymptotic couple.
A general intuition about asymptotic couples can be summurized in the following graph of on the asymtptoc couple:
On this graph, we see that for all and we have . We also see that if is nonempty, then we have for all , and has a unique fixed point which we suggestively denote here, and sometimes call .
Definition
Then the value group of gives rise to an asymtotic couple defined as in the case of Hardy fields or Hfields, (which are particular cases of asymptotic fields). It follows from the defintion that for all nonzero , we have . In this context, or for general asymptotic couples, we define
We have in . If is a Liouvilleclosed Hardy field, then the set is downward closed (i.e. initial in ), because each derivative is a logarithmic derivative.
See [2, Sections 6.5, 9.1 and 9.2] for proofs of the following facts, some of which were already proved by Rosenlicht.
The set has at most one element, and this element equals if has a maximum.
Corollary
Such an element is called a gap, and there cannot be gaps if has a maximum. Gaps remains gaps when taking divisible hulls. We call grounded if has a maximum. We say that has asymptotic integration if .
has a gap.
is grounded.
has asymptotic integration.
Example
Note that for , the element is a gap. It is not trivial to construct other asymptotic couples with a gap, but they can be realized as nonArchimedean Hardy fields or fields of transseries.
Suppose is finitely generated as an Abelian group, or more generally that its rational rank (i.e. ) is finite, or even the rank of has a valued group. Then is grounded, since in fact is finite.
If is a Hardy field which is closed under integration. Then its asymptotic couple has asymptotic integration.
Consider normalized asymptotic couples where and , and let denote the corresponding firstorder theory.
Theorem
is divisible.
has asymptotic integration.
is initial in .
In other words, we have a model companion for . Moreover, this model companion has QE in the extended language with a unary predicate for .
This is in particular the case for Liouvilleclosed Hardy fields (or Liouvilleclosed Hfields with small derivation). In that case the group is naturally an ordered vector space over , using and to define real powers
of strictly positive elements. We also have a QE result for two sorted structure expanded with this scalar multiplication. Moreover, this last structure is interpretable in the ambient Hardy field / Hfield. One interprets using the differential equation
whose ambiguities are absorbed by the valuation.
(continued in 0301: Lecture 12) Extending a Hardy field means in particular realizing cuts in . We will focus on cuts in for convenience, i.e. on subsets of without supremum in . We assume that . There are five particularly important cuts:
Symbol  Definition  Realization  In  In 
,  ,  
Note that all those cuts are definable in a uniform way in . We write to mean that the cut is realized in , meaning that there is an with . We write to mean that is realized in a Hardy field extension of but not in . We say that is free if . Likewise is free if .
For the lambda cut and the omega cut, we have other explicit quasiquadratic definitions, assuming that is also ungrounded (so not ). In particular, the field is free if and only if
For any differential field , we have a function
on . If is an Hfield, then is strictly increasing on .
As long as is ungrounded, there is a sequence which is strictly decreasing, coinitial in and satisfies whenever . The ordinal is an infinite limit ordinal. In transseries, we must have and we can take for all . Then a realization of is an Hardian with for all . This does not depend on the choice of . We write as in the finite case. Writing , then a realization of is an Hardian pseudo limit of . Note that for such a realization , the germ realizes . So freeness implies freeness (but not the other way around). Writing , we have a strictly increasing pseudoCauchy sequence , and is free if and only if has no pseudolimit in . Likewise freeness implies freeness.
Question
(continued in 0303: Lecture 13) In order to make sense of the cut and the cut, it is necessary to consider a firstorder generalization of Hardy fields, i.e. the notion of Hfields (more precisely those with constant field ).
Definition
We say that has small derivation if moreover
We'll take most our Hfields to have small derivation. In particular, each Hfield is an asymptotic field whose asymptotic couple is asymptotic.
Let us fix an ungrounded Hfield , let denote its asymptotic couple. Then for , we have
This last ambiguity cannot occur in Hardy fields, since in those the relation is determined.
Assume that is realclosed, and let . Then realizes in if and only if there is an Hfield extension of and a which realizes in with (in fact any such will realize ). Similarly, an element realizes in if and only if there is an Hfield extension of and a which realizes in with . Now consider an ungrounded Hardy field . Then for , we have
If is a realclosed Hfield, then realizes if and only if some Hfield extension has two linearly independent solutions of , and there is a differential field extension of which cannot be ordered to make it an Hfield extension of .
There are other nice consequences of freeness for realclosed Hfields with small derivation: that differentially algebraic Hfield extensions remain free, with mutually coinitial positive psisets.
Let be a differential field with (e.g. is a Liouvilleclosed Hardy field). Consider a homogeneous linear ODE
(4.3.1) 
Let with , and set . Then (4.3.1) is equivalent to
for a certain . For , we have
So has a nontrivial solution in if and only if . More details in [2, Section 5.2].
An Hfield with asymptotic integration is free if and only if it satifsifes
(4.3.2) 
Theorem
0803: Lecture 14:
Let be a differential ring (we impose in particular that is a subring of ). We write for the ring of differential polynomials with one indeterminate . As a ring this is , but it is also an extension of where extends uniquely to by setting for all . For any differential ring extension of and , we have an evaluation map
which is the unique extension of differential rings sending to . We write for the differential ring generated by over , which is the range of the above map.
Inversely, each can be seen as an operator which we often identify with .
The order of , the order of is the least with . So differential polynomials of order are just polynomials in . Any is the sum of its homogeneous parts where is the homogeneous part of of degree . The degree part is particularly important.
The ring is the ring of linear differential operators over , which in general is noncommutative. It is free as a left module with basis . The product is given by composition of operators.
The product is given by extending the rules
for and . We see that is commutative if and only if the derivation on is trivial. We also define if , and we define the order of to be . Each element of acts as a linear operator on each differential ring extension of , where is the constant ring of . Composition of operators coincides with the product. In other words, we have an embedding .
We now assume that is a differential field, with field of constants . Then has excellent algebraic properties: it is Euclidean in the expected sense (on the left and on the right).
For , the space is a finite dimensional subspace of the linear space . In fact .
Factorization in . Let . Then for in a differential field extension of , we have if and only if . So we have if and only if we have for some in some extension of with . Call irreducible if and for all of order . The Euclidean algorithm lets us write every as a product of irreducibles. The factorization is not unique (even up to units). We say that splits (over ) if for some , and . If and , then splits over if and only if both and do.
Returning to , we define additive and multiplicative conjugation. For and , we define
The order and degree are preserved by those operations, (if for multiplicative conjugation). We also have , so additive conjugation commutes with homogeneous parts.
We still work within a differential field . Let . We consider the derivation on . Rewriting in terms of can sometimes drastically simplify things.
Let denote the differential field . So and for all . For , we claim that we have a differential polynomial such that for all in and respectively (or even extensions thereof). Indeed, consider as the element of . We then have
and so on. We obtain where each lies in .
Using those identities, we define to be the unique algebra endomorphism of which sends each to itself, to itself, and each to . Note that this operation is bijective, with inverse . Indeed, we have for all and . The compositional conjugation also preserves the degree and order, and commutes with homogeneous part and additive and multiplicative conjugations.
Now let be an Hfield, with constant field . We write and . We also assume that has asymptotic integration. We consider the asymptotic couple of . For , the field is still an Hfield. The asymptotic couple of is , so its representation is just a vertical shift of that of . The field has small derivation if and only if for some , i.e for some . When using compositional conjugations, we will only consider such 's which satisfy this property, which we call active.
Important phenomenon. Let . As increases, various quantities associated to stabilize. One such quantity is the socalled dominant degree of .
Assume now that has small derivation. Then and are differential subrings of (except for the fact that does not contain ), so we have a natural differential ring mohphism with kernel , sending each to itself and taking to its residue in .
For , take an with where . Then where is the image of under . The number
does not depend on the choice of (and thus of ). We call it the dominant degree of . We tend to take with valuation where . We define the Newton degree of has the eventual value of for active with sufficiently large valuation. We also set . In fact even eventually stabilizes to a polynomial called the Newton polynomial of .
Definition
Theorem
Theorem
Theorem
The extension is unique up to isomorphism over . Besides, it is an immediate algebraic extension of .
Theorem A. The theory of free, Newtonian, Liouvilleclosed fields with small derivation is complete.
The theory of free, Newtonian, Liouvilleclosed fields is model complete and has two model completions (small der and not small der). This is the model companion of the theory of fields.
Theorem B. The field of transseries is a model of . Also the field of differentialy algebraic transseries is a model of .
Call a Hardy field maximal if it has no proper algebraic extension which is a Hardy field. In particular, maximal Hardy fields are maximal. A Hardy field is said Hclosed if it is a model of , i.e. if it is free, Newtonian and Liouvilleclosed.
Theorem
Let be a Hardy field with asymptotic integration, and let be active. Then is not a Hardy field in general, but it is isomorphic as an ordered, valued differential field to the Hardy field for any Hardian germ with . Indeed, for , we have , so
is the desired isomorphism.
Example
For , define to be the linear functions . We simply write . Given , we want to invert the linear operator
i.e. find a right inverse for . We have
where for all .
If on , then has good properties. Indeed, consider the space
This is a Banach space with norm where is the sup norm on .
Proposition
The case when is called the attractive case. The opposite, “repulsive” case, i.e. when for some on , then we need another right inverse
for the same as before. In that case, that is continuous.
We next need to consider the case which is neither attractive nor repulsive. Let be a unit. Then and can sometimes be chosen so that become attractive or repulsive. We then use compositional conjugation to work in something that is isomorphic to a Hardy field. Indeed for all units , we have
where choosing large enough, the function is in the attractive or repulsive case. The same works for complexifications, taking real parts for repulsive and attractive conditions.
Remainder about smoothness of solutions of ODE's. For of order where is a differential ring, define
Note that . Recall that is a differential subring of , and that is a differential subring of . Let have order and let , so that is defined. Suppose that . If , then . Similar results hold if is replaced by or , or even in their complexifications.
A relevant special case: assume that is a Hardy field and let be linear of order where . If is such that and that , then so , whence . The same holds in the complexification.
Consider and the operator
Assume that splits as a composition
and we have for each factor a continuous right inverse . Then we have a continuous right inverse for .
We can now explain the proof of Theorem 5.1.1. We will call a Hardy field Newtonian if it is “Newtonian for differential polynomials of order ”.
Very brief sketch of proof of Theorem 5.1.1. We want to be able to construct a algebraic Hclosed Hardy field extension of . We initially algebraically extend into a Liouvilleclosed and free Hardy field containing , and closed under . By Zorn's lemma, it is enough to show that assuming that is not Newtonian, it has a proper algebraic extension. By nonNewtonianity, there is a witness where , and is a zero of which lies in an immediate Hfield extension of , with and . We can choose this tuple to be lexicographically minimal for . Since is realclosed, we have , and is Newtonian. It follows that is not the zero of a differential polynomial of order over , so the extension is isomorphic over to via
We can also change without modifying the degrees to arrange that , so : do this by taking .
It is enough to find a germ which is Hardian such that is isomorphic over , just as a field, to . At a minimum, we want such that and . To that end, we use a fixed point construction. Let be the linear differential operator corresponding to the homogeneous degree part of . One can arrange that , and thus , have order . In order to make this sketch of proof possible, we make the bold assumption that splits over , i.e. that
for and . By using other conjugations and tricks, we can arrange that and that . Pick representatives for each germ , for a suitable common . In fact since , we can impose that by choosing large enough. Choosing even larger enougher, we impose that each factor is either in the attractive or repulsive case as per Section 5.1.2. This gives us a “good” right inverse of the geometric realization of . Consider the (nonlinear in general) operator
and note that any fixed point of is a zero of . Indeed, assume that . Then applying on both sides of the equality gives
hence the result. Recall that is a Banach space, so it is enough in order to prove that such a fixed point exists, to show that is contractive on say . This can be done after transforming into a “split normal form” through successive additive, multiplicative, compositional conjugations. More precisely, we arrange that where is “tiny” compared to both and . We have the liberty of increasing without changing the problem (e.g. still holds), and then is contractive and has fixed point which is actually infinitesimal, not in , and also lies in by Remark 5.1.3.
We note three problematic issues:
The bold assumption might fail, and it is necessary to work over in general (which is a valued field). We also need to assume that is closed under and . All of this implies that we can assume that is Newtonian
5.1.1. this notion can be made sense of in the general, nonordered context of valued fields or specifications thereof
Then one must find a way to get back into the real valued case, starting from the solution . Indeed write where and one of or is not in .
Even if one gets with and , one still needs to show that is Hardian.
As a final comment, the proof can be carried out in instead of .
For the next few lines, we will focus on some aspects related to points and above, regarding exponential sums.
Where being uniformly distributed mod means that
for each .
Functions in the linear span of are called “amlost periodic”. Using this and Boshernitzan's result, we can derive the following corrolary:
Corollary
Then for all , the germ of is infinitesimal if and only if are infinitesimal.
Now let be a Liouvilleclosed Hardy field containing , and assume that is linearly surjective, i.e. all linear differential equations of order over have solutions in . For where , we have where by linear surjectivity. So . Note that .
Write
So is a differential subring of extending and containing all constants (i.e. complex numbers). We call the universal exponential extension of . For , we have and . In fact . If moreover , then already, so we are more interested in 's which are positive infinite. We fix a decomposition into linear spaces (think of as a space of purely infinite series). Note that is an isomorphism. We have that . This gives the structure of a group ring over , with as the group.
Proposition
Proof. This family clearly generates over . Now assume for contradiction that for some , and . Then Corollary 5.2.1 implies that are infinitesimal. But multipliying by a large , we can assume that at least one is not infinitesimal: a contradiction.
Corollary
Below, let range in . When using expressions like , we always assume that the family has finite support. Note that for , we have , whence
In particular, a basis of as an vector space is given by . We extend the valuation on to by setting
This is the unique valuation on which extends the valuation on . We have a corresponding dominance relation denoted on . Note that for , we have if and only if for all , whence, by Corollary 5.2.1, if and only if in . In fact:
Proposition
Let us show that each acts on in a very transparent way. Let . Then one can see that
Thus solving in reduces to solving systems of equations in .
Proposition
and for any such basis the 's with form a basis of . If morover splits over , then , whence .
Corollary
where , and . For any such basis, the family is a basis of . If moreover splits over , then , whence .
We now look into the proof of the following theorem
Theorem
The special case was already treated by
Hausdorff called a linear ordering an set if for all countable subsets with , there is an with . So Theorem 6.0.1 is equivalent to the following:
Corollary
Corollary
Corollary
It is unknown whether Theorem 6.0.1 holds in the analytic setting (but it does in the smooth setting).
Let us start with a valuation theoretic characterization of ness in ordered valued fields. Let be an ordered field, with its natural valuation. Recall that its residue field is Archimedean, hence it embeds uniquely into .
A sequence in is said to be pseudoCauchy (pc for short) if there exists a such that for all with , we have . Equivalently, we have for all .
Let be an ordered field extension of . An element is a pseudolimit of a sequence if there is an such that for all , we have . Note that this implies in particular that is pseudoCauchy. We then say that pseudoconverges (to ) and we write .
Example
Lemma
is .
The ordered residue field is (isomorphic to) , every pcsequence indexed by has a pseudolimit in , and the value group of is as an ordered set.
For (maximal) Hardy fields, the first part is a given, but maybe not the other two... We will focus on the pcsequence part of the work.
Before we start, let us reformulate the problem of finding pseudolimits in and out of ordered fields. Let be a pseudoCauchy sequence in . All subsequences of are aso pseudoCauchy, and share limits in all ordered (and naturally valued) field extensions of . By passing to a subsequence, we can arrange that is strictly monotonous, and given our ness problem, we might as well take opposites in the strictly decreasing case, hence imposing that is strictly increasing. Similarly, we may assume by translation and by taking a final segment of that: each is strictly positive (in particular ), and that for all . Now for such a sequence , we define and for all . Then we have
(6.1.1) 
and for all . Using this, one can show that the following are equivalent:
All pcsequences in pseudoconverge in .
For all satisfying (6.1.1), the pcsequence pseudoconverges in .
Let be a Hausdorff field containing and let be strictly positive elements of with . Set for each . Let us try to construct a Hausdorff field which contains a pseudolimit of . Pick, by induction on , a continuous representative of each germ , such that and for all . Thus, for each , the sum is defined. The convergence of this series of functions is uniform on compact subsets of , hence is actually continuous on .
Exercise
Lemma
[missing notes here, we take back after the proof of the main filling cuts result in the fluent case].
Assume now that is cofinal in . In particular, the cofinality of the psi set is . Let be a sequence of positive active elements in such that is strictly increasing and cofinal in .
Remark
(6.3.1) 
However, this depends on . In particular, for each , writing for the derivation on , we obtain a satisfying (6.3.1) above, with respect to .
Let us construct a partition of unity such that exists and satisfies for all . Then one can show that for all active in and all , we have
Then the key lemma implies that is an Hardian pseudolimit of . We choose as smooth functions that are zero outside of an interval , one on increasing on , decreasing on , and with on . In fact we have a pointwise sum everywhere.
Conjecture
Question
We now turn to the second part of the proof where we want to prove that the (underlying ordering of the) value group of a maximal Hardy field is an set.
Theorem
, and , i.e. generates a countable cut in .
there are sequences and in and respectively such that is linearly independent modulo , where , for all . So .
.
Then there is an Hardian germ such that realizes the same cut as in .
Remark
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active element 23
asymptotic couple 17
asymptotic integration 18
dominant degree 24
field 20
gap 18
grounded asymptotic couple 18
Hardian over 14
Hardian 14
maximal Hardy field 25
Newton degree 24
Newton polynomial 24
Newtonian Hfield 24
order of a differential polynomial 22
pseudoCauchy sequence 31
pseudolimit 31
small derivation 21