An Intermediate Value Property for FirstOrder Differential
Polynomials
Lou van den Dries
University of Illinois
Algorithms Seminar
June 28, 1999
[summary by Philippe Dumas & Bruno Salvy]
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A theorem of Rubel [4] shows that solutions of algebraic
differential equations can present pathological asymptotic
behaviours. Therefore when studying differential equations from an
asymptotic viewpoint it is natural to restrict to solutions obeying
extra smoothness conditions. A convenient context for these questions
is provided by Hardy fields, which are defined below. A typical
differential equation dealt with by the techniques of this work is
F(x,y,y')=(x+y^{2})y'^{3}e^{x}+yy'log 
x+y^{4}e 

=0. 
It is easily seen that for y=e^{e}^{x}, F(x,y,y') is asymptotically
positive, while for y=x, F(x,y,y') is asymptotically negative. It
is therefore natural to wonder whether there exists a solution to this
equation whose growth is between that of x and
of e^{e}^{x}. The work summarized here [5] gives a positive
answer to this question, and proves that there exists such a solution in a
Hardy field.
1 Hardy Fields
A Hardy field is a field closed under differentiation, whose
elements are germs at ¥ of realvalued
functions [1]. (Think of it as the set of possible
asymptotic behaviours.)
Examples of Hardy fields are the field R of (germs of) constant
functions, the field R(x) of (germs of) rational functions
over R. Hardy fields are named after G. H. Hardy, who proved
in [2] that explog functions (i.e., functions
obtained from R(x) by field operations, the functions exp and
log.) form a Hardy field.
The main constraint here is that nonzero elements of Hardy fields
have to be invertible, and thus cannot have arbitrarily large
zeros. Consequently, since their derivatives belong to the field,
they have to be ultimately monotonic and tend to a possibly
infinite limit. Also, differences of two (germs of) functions of a
Hardy field are also in the field and possess a limit, so that this
field is ordered. A Hardy field K can be extended by a C^{¥}
function y if
for all polynomials PÎK[Y], P(y) is either 0 or does not have
arbitrarily large zeros. A Hardy field K can be extended by
real solutions of polynomials in K[Y] and by antiderivatives of
elements of K [3]. This is how explog
functions can be built from R(x).
The order induces a natural topology, a basis of the open sets being
given by the open intervals. Thus
continuous functions are defined and for instance, the
differentiation operator is continuous since y'>f
implies y>òf. This is an open set since the field can be
extended by òf if necessary.
The aim of this work is to prove the following.
Theorem 1 Let K be a Hardy field and FÎK[x,y]. Assume
there exist f and y in K such that F(f,f')<0<F(y,y').
Then there exists h in a Hardy field extension of K such
that F(h,h')=0.
The function y® F(y,y') is continuous, but in general Hardy
fields are not Archimedean (consider 1 and x
in R(x)). Consequently, the intermediate value theorem may not hold.
The proof consists in lifting properties of continuous functions
over R to Hardy fields. The same question for higherorder
differential polynomials is still a conjecture.
2 Basic Case
We first consider equations of the form
where G is C^{1} in the neighborhood of S={(x,y),r£ x,a(x)£ y£ b(x)},
for some rÎR and a and b in a Hardy field K giving
different signs to y'(x)G(x,y(x)).
A simple reasoning based on the intermediate value theorem
shows the existence of a C^{1} solution h(x)
of (1) with a(x)<h(x)<b(x) for x³ r.
If, moreover, x® G(x,h(x)) belongs to K for all h in K,
then h belongs to an extension of K. This is proved in three
steps. First, if h has arbitrarily large zeros then so
does x® G(x,0) but since this belongs to a Hardy field, we
get G(x,0)=0 and h=0 is the unique solution of the differential
equation. Next, if h¹h belongs to an extension of K,
then q=hh cannot have arbitrarily large zeros, using the same
argument as before with the equation
q'(x)=G(x,q(x)+h(x))h'(x).
Noting that any polynomial PÎK[Y] can
be factored in linear factors or quadratic factors with negative
discriminant extends this argument to P(h) and concludes the proof.
3 General First Order Case
The aim is to reduce the general case F(y,y')=0 to the basic case
considered in the previous section. This is achieved through an
analogue of the cylindricalgebraic decomposition: the
interval (f,y) is split into subintervals
f=a_{1}<...<a_{n}=y
such that in every interval (a_{i},a_{i+1}) there are finitely many
functions f_{i,j} algebraic over K and the polynomial F(y,z) has
constant sign in the cell a_{i}<y<a_{i+1},
f_{i,j}(y)<z<f_{i,j+1}(y). Note that everything here also depends
on x through the coefficients of F.
It is now sufficient to exhibit two functions a,b,
with a_{i}<a<b<a_{i+1} for some i, such
that a'f_{i,j}(a)<0<b'f_{i,j}(b) for some j.
Let A be the set of yÎ(f,y) such that F(y,y')<0 and
similarly B for F(y,y')>0. If A (resp. B) has an upper
(resp. lower) bound, then this is a solution of the equation and we
are done. Otherwise, it is possible to select aÎ A and bÎ B
belonging to the same interval (a_{i},a_{i+1}) (if not, one of
the a_{i}'s would be a solution of the equation). Necessarily,
(a,a') and (b,b') do not belong to the same cell and therefore one
of the f_{i,j} fulfills our needs. We denote it f.
The reduction to the basic case requires that the
application (x,y)® f(y) be C^{1} in the domain S.
This follows from the analyticity of the
roots of a polynomial equation with respect to its coefficients
outside of singular varieties and the C^{1} property of these
coefficients for x sufficiently large.
References
 [1]

Bourbaki (N.). 
Éléments de Mathématiques, Chapter V: Fonctions
d'une variable réelle (appendice), pp. 3655. 
Hermann, Paris, 1961, 2nd edition.
 [2]

Hardy (G. H.). 
Orders of Infinity. 
Cambridge University Press, 1910, Cambridge
Tracts in Mathematics, vol. 12.
 [3]

Rosenlicht (Maxwell). 
Hardy fields. Journal of Mathematical Analysis and
Applications, vol. 93, n°2, 1983, pp. 297311.
 [4]

Rubel (L. A.). 
A universal differential equation. Bulletin of the American
Mathematical Society, vol. 4, n°3, May 1981,
pp. 345349.
 [5]

van den Dries (Lou). 
An intermediate value property for firstorder differential
polynomials. 
1999. Preprint.
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.