|
Abstract
Let 
be the field of grid-based
transseries or the field of transseries with finite logarithmic
depths. In our PhD. we announced that given a differential polynomial
with coefficients in
and transseries 
with 
and 
, there exists an 
, such that 
. In this note, we will prove this
theorem.
Let 
be a totally ordered exp-log field. In
chapter 2 of [vdH97], we introduced the field 
of transseries in 
of finite
logarithmic and exponential depths. In chapter 5, we then gave an (at
least theoretical) algorithm to solve algebraic differential equations
with coefficients in 
. By
that time, the following theorem was already known to us (and stated in
the conclusion), but due to lack of time, we had not been able to
include the proof.
Theorem be a differential polynomial with coefficients in
. Given
in , such that 
, there exists an with
.
In the theorem, 
stands for the open interval
between 
and 
.
The proof that we will present in this note will be based on the
differential Newton polygon method as described in chapter 5 of [vdH97]. We will freely use any results from there. We recall
(and renew) some notations in section 2.
In chapter 1 of [vdH97], we also introduced the field of
grid-based 
transseries in . In chapter 12, we have shown that our algorithm
for solving algebraic differential equations preserves the grid-based
property. Therefore, it is easily checked that theorem 1
also holds for 
. Similarly,
it may be checked that the theorem holds if we take for
the field of transseries of finite logarithmic depths (and possibly
countable exponential depths).
Assume that 
is a differential polynomial with
coefficients in , which
admits a sign change on a non empty interval of
transseries. The idea behind the proof of theorem 1 is very
simple: using the differential Newton polygon method, we shrink the
interval further and further while preserving
the sign change property. Ultimately, we end up with an interval which
is reduced to a point, which will then be seen to be a zero of .
However, in order to apply the above idea, we will need to allow non
standard intervals in the proof. More precisely,
and may generally be
taken in the compactification of ,
as constructed in section 2.6 of [vdH97]. In this paper we
will consider non standard (resp. ) of the following forms:
, with 
;
, with ;
, with
and where 
is a transmonomial.
, with
and where is a transmonomial.
, with
and 
.
Here 
and 
respectively
designate the infinitely small and large constants 
and 
in the compactification of . Similarly, 
and 
designate the infinitely small and large constants
and 
in the
compactification of . We may
then interpret as a cut of the transline into two pieces 
.
Notice that
Remark 
, 
, and so on are redundant. Indeed,
does not depend on 
, we have
whenever 
,
etc.
Now consider a generalized interval 
,
where and may be as
above. We have to give a precise meaning to the statement that admits a sign change on 
.
This will be the main object of sections 3 and 4.
We will show there that, given a cut of the
above type, the function 
may be prolongated by
continuity into from at least one direction:
If 
, then 
is constant on 
for some 
.
If 
, then is constant on 
for some 
.
If 
, then is constant on for some .
If 
, then is constant on 
for some .
If 
, then is constant on for some .
(In the cases 
, 
and so on, one has to interchange left and right
continuity in the above list.) Now we understand that
admits a sign change on a generalized interval
if 
.
. |
(1) |
Here we use vector notation for tuples 
and 
of integers:
along orders. |
(2) |
In this notation, 
runs through tuples 
of integers in 
of length 
at most 
, and
for all permutations of integers. We again use
vector notation for such tuples
We call 
|| the weight
of and
the weight of .
Additive conjugation:
 |
(3) |
Multiplicative conjugation:
 |
(4) |
Upward shifting (compositional conjugation):
 |
(5) |
where the 
are generalized Stirling numbers of
the first kind:
near zero and infinity near
infinity
Lemma be a differential polynomial with coefficients in
. Then 
has constant sign for all sufficiently large 
.
Proof. If 
,
then the lemma is clear, so assume that 
.
Using the rules
we may rewrite 
as an expression of the form
 |
(6) |
where 
and 
for each 
. Now consider the lexicographical
ordering 
on 
,
defined by
This ordering is total, so there exists a maximal
for , such that 
. Now let 
be
sufficiently large such that 
for all 
. Then
 |
(7) |
for all postive, infinitely large 
,
since 
for all such 
.
near
zero
Lemma be a differential polynomial with coefficients in
. Then 
has constant sign for all sufficiently small 
.
Proof. If ,
then the lemma is clear. Assume that and rewrite
as in (6). Now consider the twisted
lexicographical ordering 
on , defined by
This ordering is total, so there exists a maximal
for , such that . If is sufficiently
large such that for all , then
 |
(8) |
for all postive infinitesimal 
.
Assume that has purely exponential coefficients.
In what follows, we will denote by 
the purely
exponential differential Newton polynomial associated to a monomial
, i.e.
 |
(9) |
where
 |
(10) |
The following theorem shows how 
looks like after
sufficiently many upward shiftings:
Theorem be a differential polynomial with purely exponential
coefficients. Then there exists a polynomial 
and
an integer 
, such that for
all 
, we have 
.
Proof. Let be minimal, such
that there exists an 
with 
and 
. Then we have 
and
 |
(11) |
by formula (5). Since 
,
we must have 
. Consequently,
. Hence, for some 
, we have 
. But then (11) applied on 
instead of yields 
. This shows that 
is
independent of 
, for .
In order to prove the theorem, it now suffices to show that 
implies 
for some polynomial . For all differential polynomials
of homogeneous weight , let
 |
(12) |
Since 
, it suffices to show
that whenever 
.
Now implies that 
.
Furthermore, (5) yields
 |
(13) |
Consequently, we also have 
.
By induction, it follows that 
for any iterated
exponential of 
. We conclude
that 
, by the lemma 3.
Remark with coefficients in , this polynomial becomes purely exponential
after sufficiently many upward shiftings. After at most 
more upward shiftings, the purely exponential Newton polynomial
stabilizes. The resulting purely exponential differential Newton
polynomial, which is in 
, is
called the differential Newton polynomial of .
near constants
In the previous section, we have seen how to compute 
and 
for all .
In this section, we show how to compute 
and 
for all and all
transmonomials . Modulo an
additive and a multiplicative conjugation with 
resp. , we may assume without
loss of generality that 
and 
. Hence it will suffice to study the behaviour
of 
for 
and positive
infinitesimal (but sufficiently large) 
,
as well as the behaviour of 
for positive
infinitely large (but sufficiently small) 
.
Modulo suffiently upward shiftings (we have 
and
), we may assume that has purely exponential coefficients. By theorem 5 and modulo at most 
more upward
shiftings, we may also assume that
 |
(14) |
for some polynomial and 
. We will denote by 
the
multiplicity of 
as a root of 
. Finally, modulo division of
by its dominant monomial (this does not alter ), we may assume without loss of generality that
.
in
between constants
Lemma 
with 
,
the signs of 
and 
are
independent of 
and given by
 |
(15) |
Proof. Since is purely
exponential and , there
exists an 
such that
 |
(16) |
for all 
. Let 
be such that 
,
where 
. Then 
, whence
 |
(17) |
Furthermore, 
, whence
 |
(18) |
Put together, (17) and (18) imply that 
. Hence 
, by (16). Now
 |
(19) |
since 
for all positive infinitesimal .
Corollary is homogeneous of degree ,
then
 |
(20) |
for all with .
Corollary 
be constants such that 
. Then there exists a constant 
with 
.
Proof. In the case when is odd,
then 
holds for any 
with
, by (15).
Assume therefore that is even and let 
denote the multiplicities of 
as
roots of 
. From (15)
we deduce that
 |
(21) |
In other words, the signs of 
for 
and 
are different. Hence, there
exists a root 
of between
and 
which has odd
multiplicity 
. For this root
, (15) again
implies that .
before
and after the constants
Lemma 
with 
,
the signs of 
and are
independent of and given by
 |
(22) |
Proof. Since is purely
exponential and , there
exists an such that
 |
(23) |
since 
. Furthermore 
and 
, whence
. In particular, 
, so that 
,
by (23). Now
 |
(24) |
since 
for positive infinitely large .
Corollary is homogeneous of degree ,
then
 |
(25) |
for all with .
Corollary be a constant such that 
. Then there exists a constant
with .
Proof. In the case when is odd,
then holds for any with
, by (15).
Assume therefore that is even and let 
be the multiplicity of as a root
of . From (15)
and (22) we deduce that
 |
(26) |
In other words, the signs of for and 
are different. Hence, there
exists a root of which
has odd multiplicity . For
this root , (15)
implies that .
It is convenient to prove the following generalizations of theorem 1.
Theorem and 
be a transseries
resp. a transmonomial in .
Assume that changes sign on an open interval
of one of the following forms:
, for some 
with 
.
.
.
.
Then changes sign at some 
.
Theorem and 
be a transseries
resp. a transmonomial in .
Assume that changes sign on an open interval
of one of the following forms:
, for some with .
.
.
.
Then changes sign on 
for some with 
.
Proof. Let us first show that cases a, b
and d may all be reduced to case c. We will show this in
the case of theorem 13; the proof is similar in the case of
theorem 14. Let us first show that case a may be
reduced to cases b, c and d. Indeed, if changes sign on 
,
then changes sign on 
, 
or 
. In the second case, modulo a multiplicative
conjugation and upward shifting, corollary 9 implies that
there exists a 
such that
admits a sign change on 
.
Similarly, case d may be reduced to cases b and c
by splitting the interval in two parts. Finally, cases b and
c are symmetric when replacing by .
Without loss of generality we may assume that 
, modulo an additive conjugation of
by . We prove the theorem by
a triple induction over the order 
of , the Newton degree 
of the asymptotic algebraic differential equation
 |
(27) |
and the maximal length 
of a sequence of
privileged refinements of Newton degree (we have
, by proposition 5.12 in [vdH97]).
Let us show that, modulo upward shiftings, we may assume without loss of
generality that and are
purely exponential and that 
.
In the case of theorem 13, we indeed have 
and 
. In the case of theorem
14, we also have 
.
Furthermore, if 
is such that 
changes sign on 
, then 
is such that changes sign on
.
Case 1: (27) is quasi-linear. Let 
be the potential dominant monomial relative to (27). We may
assume without loss of generality that 
,
modulo a multiplicative conjugation with .
Since By , we have 
or 
for certain constants 
.
In the case when , there
exists a solution to (27) with 
. Now 
and 
. We claim that 
and
must be equal. Otherwise 
would admit a solution between 
and 
, by the induction hypothesis. But then the
potential dominant monomial relative to (27) should have
been 
, if 
is the largest such solution. Our claim implies that 
, so that 
.
Finally, lemma 4 implies that
admits a sign-change at .
Lemma 7 also shows that 
.
In the case when , then any
constant 
is a root of 
. Hence, for each 
,
there exists a solution to (27)
with 
. Again by lemmas 4 and 7, it follows that
admits a sign change at and on .
Case 2: 
. Let be the largest classical potential dominant monomial
relative to (27). Since 
(resp.
), one of the following
always holds:
We have 
(resp. 
).
We have 
.
We have 
.
For the proof of theorem 14, we also assume that 
in the above three cases and distinguish a last case
2d in which 
.
Case 2a. We are directly done by the induction hypothesis, since the equation
 |
(28) |
has a strictly smaller Newton degree than (27).
Case 2b. Modulo multiplicative conjugation with , we may assume without loss of generality that
. By corollary 12,
there exists a 
such that 
. Actually, for any transseries 
we then have 
. Take 
such that
 |
(29) |
is a privileged refinement of (27). Then either the Newton
degree of (29) is strictly less than , or the longest chain of refinements of (29)
of Newton degree is strictly less than . We conclude by the induction
hypothesis.
Case 2c. Since is the largest classical
dominant monomial relative to (27), the degree of the
Newton polynomial associated to any monomial between
and must be .
Consequently,
 |
(30) |
By the induction hypothesis, there exists a monomial 
with 
and
 |
(31) |
In other words, is a dominant monomial, such
that 
and
 |
(32) |
We conclude by the same argument as in case 2b, where we let play the role of .
Case 2d. Since is the largest classical
dominant monomial relative to (27), the degree of the
Newton polynomial associated to any monomial between
and must be .
Consequently,
 |
(33) |
By the induction hypothesis, there exists a monomial
with 
and
|
(34) |
In other words, is a dominant monomial, such
that 
and
|
(35) |
We again conclude by the same argument as in case 2b.
Corollary
admits a root in .
Proof. Let be a polynomial of
odd degree with coefficients in .
Then formula (7) shows that for sufficiently large 
we have 
,
since 
is odd in this formula. We now apply the
intermediate value theorem between 
and .