The rational points of a definable set Pila, J. and Wilkie, AJ 2006 MIMS EPrint

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The rational points of a definable set
Pila, J. and Wilkie, A. J.
2006
MIMS EPrint: 2007.198
Manchester Institute for Mathematical Sciences
School of Mathematics
The University of Manchester
Reports available from: http://eprints.maths.manchester.ac.uk/
And by contacting: The MIMS Secretary
School of Mathematics
The University of Manchester
Manchester, M13 9PL, UK
ISSN 1749-9097
2005.10.10
The rational points of a definable set
J.PilaandA.J.Wilkie
Abstract
LetX ⊂ Rnbeasetthatisdefinableinano-minimalstructureoverR. Thispaper
showsthat, inasuitablesense, thereareveryfewrationalpointsofX thatdonot
lieonsomeconnectedsemialgebraicsubsetofX ofpositivedimension.
2000MathematicsSubjectClassification: 11G99,03C64
1. Introduction
Thispaperisconcernedwiththedistributionofrationalandintegerpointsoncertainnon-
algebraic sets in Rn. To contextualize the kind of results sought, and in particular to motivate
the present setting of definable sets in o-minimal structures over R (see 1.7 below), we begin
bydescribingearlierresults.
The ideas pursued here grew from the paper [4] of Bombieri and the first author, where
a technique using elementary real-variable methods and elementary algebraic geometry was
used to establish upper bounds for the number of integer points on the graphs of functions
y = f(x),undervariousnaturalsmoothnessandconvexityhypotheses. Resultswereobtained
forf variouslyassumedtobe(sufficiently)smooth,algebraic, orreal-analytic. Severalresults
concernedthehomotheticdilationofafixedX : y = f(x).
1.1. Definition. LetX ⊂ Rn. Forarealnumbert ≥ 1(whichwillalwaysbetacitlyassumed),
the homothetic dilation of X by t is the set tX = {(cid:3)tx ,...,tx (cid:4) : (cid:3)x ,...,x (cid:4) ∈ X}. By
1 n 1 n
X(Z)wedenotethesubsetofX comprisingthepointswithintegercoordinates.
Suppose now that X is the graph of a function f : [0,1] → R. Trivially, one has
#(tX)(Z) ≤ t+1(withequalityfore.g. f(x) = xandpositiveintegralt). AccordingtoJarnik
[14], a strictly convex arc Γ : y = g(x) of length (cid:1) contains at most 3(4π)−1/3(cid:1)2/3 +O((cid:1)1/3)
integer points (and moreover the exponent and constant are best possible). So if X is strictly
convex,oneinfersthat
#(tX)(Z) ≤ c(X)t2/3.
However, Swinnerton-Dyer [31] showed that a substantially better estimate may be ob-
tainediff isassumedC3 andstrictlyconvex,namely
#(tX)(Z) ≤ c(X,(cid:3))t3/5+(cid:1)
forallpositive(cid:3). (Regardingthiscircleof“limited smoothness”problemsseealso[29,20].)
CountingintegerpointsontX isofcoursethesameascountingpoints(cid:3)m/t,n/t(cid:4)onX,
andinthisguisesuchquestionsaroseinworkofSarnak[27]onbettinumbersofabeliancovers.
∞
Heconjectured thatiff isC andstrictlyconvextheninfact
#(tX)(Z) ≤ c(X,(cid:3))t1/2+(cid:1)
forallpositive(cid:3). Theexponent1/2isbestpossiblehereinviewoff(x) = x2. Thisconjecture
wasthestartingpointof[4],whereitisaffirmed.
1
If f is assumed transcendental analytic, however, then the exponent may be reduced to (cid:3),
andthisresultof[4]istheprototypefortheresultstobepresentedhere.
1.2. Theorem. ([4,Theorem1])Letf : [0,1] → Rbeatranscendentalreal-analyticfunction.
LetX bethegraphoff,andlet(cid:3) > 0. Thereisaconstantc(X,(cid:3))suchthat
#(tX)(Z) ≤ c(X,(cid:3))t(cid:1).
Theorem1.2answeredaquestionalsoraisedbySarnak[27],andhasfoundapplicationin
that context [19, 28]. It can be adapted to a result on rational points of bounded height, which
weproceedtostate.
1.3. Definition. Let H : Q → R denote the usual height function: H(a/b) = max(|a|,b)
where a,b ∈ Z,b > 0,gcd(a,b) = 1. Extend to H : Qn → R by setting H(cid:3)α ,...,α (cid:4) =
1 n
max(H(α )). IfX ⊂ Rn,letX(Q)denotethesubsetofpointswithrational coordinates. Set
j
(forT ≥ 1)
X(Q,T) = {P ∈ X(Q),H(P) ≤ T}
anddefinethedensityfunctionofX tobe
N(X,T) = #X(Q,T).
ThisisnottheusualprojectiveheightH ,whichisnotespeciallynaturalinthecontext
proj
of non-algebraic sets, although this makes no difference to our results. The density function
N(X,T) is a natural function to study in situations where X(Q) may be infinite, and is the
primary object of study here. The aim is to establish upper bounds for N(X,T) under natural
geometric conditions on X, and with the guiding idea that transcendental sets should contain
“few”rationalpoints,inanappropriatesense.
1.4. Theorem. ([20,Theorem9])Letf : [0,1] → Rbeatranscendentalanalyticfunctionwith
graphX and(cid:3) > 0. Thereisaconstantc(X,(cid:3))suchthat
N(X,T) ≤ c(X,(cid:3))T(cid:1).
Nowiff isofspecialform(e.g. f(x) = ex,or,say,aG-function)thenonemayhavemuch
stronger results (or at least conjectures) on the scarcity of rational (or even algebraic) points
(seee.g. [1]). Attheotherextreme,constructionsgoingbacktoWeierstrass(seee.g. [18,25])
showthatanentiretranscendentalf maytakerationalvaluesateveryrationalargument. These
constructionsdonottakemuchcareoftheheightdensityofpoints.
However,givenanyfunction(cid:3) : [1,∞) → R,strictlydecreasingwith(cid:3)(t) → 0ast → ∞,
it is possible (see [22, 7.5]) to construct a transcendental analytic function f on [0,1] and a
(ratherlacunary)sequenceofpositiveintegersT → ∞suchthat
j
N(X,T ) ≥ T(cid:1)(Tj).
j j
Thus the above result cannot be much improved in general. (E.g. taking (cid:3)(t) = (logt)−1/2
showsthatforcertainX noboundofformN(X,T) ≤ C(logT)K holdsetc.)
2
Letusnowturntohigherdimensionalsets. Theideaofapplyingthemethodsof[4]tothe
density function of a higher dimensional transcendental analytic set — e.g. the graph X of a
transcendental analytic function f : [0,1]2 → R — was prompted by a question of Bourgain
(see [22]). Those methods entail studying the intersections of X with algebraic hypersurfaces
of high degree. Such intersections may be highly singular, though they are still semianalytic.
One is further led to consider images under projections: and thus one is led to the class of
(bounded)subanalyticsets([3])asthenatural(andmoregeneral)classofsetstoconsider. The
fundamentalpropertiesofsubanalyticsets: Uniformizationtheorem,Gabrielov’stheorem,and
Tamm’s theorem (see [3]), enabled the first author to establish analogues of 1.2 and 1.4 for
compactsubanalyticsetsofdimension2inRn.
To frame this result, it is necessary to consider a new feature that arises for sets X of
dimension2andhigher: Namely,X maycontainsemialgebraicsubsetsofpositivedimension,
evenifX itselfisnotsemialgebraic,andthesesemialgebraicsubsetsmaycontain“many”,i.e.
>> Tδ forsomepositiveδ,rationalpointsofheight≤ T. Forexample,ifX = {(cid:3)x,y,z(cid:4) : z =
xy,x,y ∈ [1,2]}theneachrationalygivesrisetoasemialgebraiccurveinX. Toaccommodate
thisfeature,wemakethefollowingdefinition.
1.5. Definition. Let X ⊂ Rn. The algebraic part of X, denoted Xalg, is the union of all
connected semialgebraic subsets of X of positive dimension. The transcendental part of X is
thecomplementX −Xalg.
Thealgebraicpartofasetmaybehardtoidentifyingeneral,anditmaybeacomplicated
set: in the above example it consists of infinitely many connected components (one for each
y ∈ Q∩[1,2]),andsoisnotsubanalytic: thispointisdiscussedfurtherbelow. However,ifthe
algebraic part is excluded from the count of rational points, a result of the quality as 1.4 may
beobtained.
1.6. Theorem ([23, Theorem 1.1]) Let X ⊂ Rn be a compact subanalytic set of dimension 2
andlet(cid:3) > 0. Thereisaconstantc(X,(cid:3))suchthat
N(X −Xalg,T) ≤ c(X,(cid:3))T(cid:1).
The corresponding result for integer points on the dilation of such a set is in [22], where
1.2and1.4arealso“upgraded”toapplytoany1dimensionalcompactsubanalyticsetX ⊂ Rn.
The exclusion of Xalg, where rational points may well accumulate, is weakly analogous
to(andsuggestedby)thenotionofthespecialsetindiophantinegeometry,andthephilosophy
of “Geometry Governs Arithmetic” [13, §F.5]. The conjecture of Lang ([16, Ch I §3; or 13,
§F.5]) asserts that an algebraic variety V is mordellic (i.e. has only finitely many rational
points) outside its special set. When V is a curve, this is the Mordell conjecture, proved by
Faltings. In higher dimensions it remains very much open. The special set is the union of
non-constantimagesofprojectivespacesandellipticcurves. Excludingthealgebraicpartfrom
ourcountismuchcoarser,butseemstobetheappropriatewayofseparatingouttheessentially
transcendental partoftheproblem.
For results on integer points on surfaces, and higher dimensional hypersurfaces, under
hypothesesofadifferential-geometric naturesee[2,29,30].
3
The generalization of 1.6 to arbitrary compact subanalytic sets in Rn was conjectured in
[22]. Such sets are globally subanalytic (meaning subanalytic as subsets of the compact real
analyticmanifoldPn(R)). Theclassofgloballysubanalyticsetsisanexampleofano-minimal
structure,andthisturnsouttobethenaturalsettingforourarguments.
Thenotionofano-minimalstructurearoseinthestudyofmodel-theoreticquestionsabout
R(“Tarski’sproblem”;[6]). Hereisadefinition(following[34];however,readerswillneedto
befamiliar withthetheoryasdevelopede.g. in[7]tofollowourproofs).
1.7. Definition. Apre-structureisasequenceS = (S : n ≥ 1)whereeachS isacollection
n n
ofsubsetsofRn. Apre-structureS iscalledastructure(overtherealfield)if,foralln,m ≥ 1,
thefollowingconditionsaresatisfied:
(1) S isabooleanalgebra(undertheusualset-theoretic operations)
n
(2) S containseverysemi-algebraic subsetofRn
n
(3) ifA ∈ S andB ∈ S thenA×B ∈ S
n m n+m
(4) ifm ≥ nandA ∈ S thenπ[A] ∈ S ,whereπ : Rm → Rn isprojectionontothefirstn
m n
coordinates
If S is a structure and X ⊂ Rn, we say X is definable in S if X ∈ S . If S is a structure and,
n
inaddition,
(5) theboundaryofeverysetinS isfinite
1
thenS iscalledano-minimal structure(overtherealfield).
The paradigm example of an o-minimal structure is provided by the semialgebraic sets:
closureunderprojectionsisthecontentoftheTarski-Seidenbergtheorem[7].
The class of semi-analytic sets is not closed under (proper) projections (see the classical
example due to Osgood in [3]). But, as mentioned above, the globally subanalytic sets form
ano-minimalstructure,denotedR : Itistheclosureundercomplementationthatishardestto
an
establish: Theabovementioned theoremofGabrielovisthekeytohisproofofthisfact[10].
Anotherexampleofano-minimalstructureisthestructureR “generatedby”ex. Here
(cid:1) (cid:2)exp
S isthecollectionofsubsetsofRn oftheformX = π f−1(0) where,forsomem ≥ n,f :
n
Rm → R is an exponential polynomial (i.e. f(x ,...,x ) = Q(x ,...,x ,ex1,...,exm)
1 m 1 m
forsomepolynomialQ ∈ R[X ,...,X ]),andwhereπ : Rm → Rn isprojectiononthefirst
1 2m
ncoordinates. Therequisitefinitenessproperty(5)followsfromKhovanskii’stheorem[15]. In
showingthatR isastructure,themajordifficultyisagainthe“theoremofthecomplement”,
exp
whichwasestablishedbythesecondauthorin[33]. NotethatR containse.g. ([9])setssuch
exp
as{(cid:3)x,xr(cid:4),x > 0}forpositiveirrationalr,or{(cid:3)x,e−1/x(cid:4),x > 0}whichdonotbelongtoR
an
(theyarenotsubanalyticattheorigin).
In fact the structure R generated by the union of R and R is o-minimal [8].
an,exp an exp
However it is known [26] that there is no “largest” o-minimal structure, and that there are
o-minimal structurescontainingnowhereanalytic functions.
Thus o-minimal structures provide rich and flexible categories to work in, while at the
sametimeanaturalsettingforourmethods.
Let then S be an o-minimal structure (over R). A definable set X ⊂ Rn will mean a set
definableinS. Nowwecanstateapreliminary versionofourmainresult.
4
1.8. Theorem. (First version) Let X ⊂ Rn be a definable set, and (cid:3) > 0. There is a constant
c(X,(cid:3))suchthat
N(X −Xalg,T) ≤ c(X,(cid:3))T(cid:1).
The proof of the theorem begins by showing that the points in question reside on “few”
(i.e. O (T(cid:1))) hypersurfaces of suitable degree d((cid:3)); it then proceeds by induction on the
X,(cid:1)
dimension of X. Thus it is necessary to have an estimate of the same form as above for those
hypersurfaceintersectionsbutinwhichtheimpliedconstantisuniformoverallintersectionsof
X withhypersurfacesoffixeddegree: i.e.,aresultforadefinablefamilyofsets. Thefollowing
convention will be adopted. In considering subsets Z = {(cid:3)x,y(cid:4)} ⊂ Rn × Rm, projection
on the first factor will be denoted π , and on the second π . Put Y = Y = π (Z) and for
1 2 Z 2
y ∈ Y put Z = {z ∈ Z : π (z) = y}, and X = X = π (Z ) its image in Rn. A family
y 2 y Z,y 1 y
Z ⊂ Rn×Rmofsetswillmeanthecollectionoffibres{X : y ∈ Y }. AfamilyZ isdefinable
y Z
ifthesetZ is. Theresulttobeprovedisthenthefollowing.
1.9. Theorem. (Secondversion)LetZ ⊂ Rn×Rm beadefinablefamily,and(cid:3) > 0. Thereis
aconstantc(Z,(cid:3))withthefollowingproperty. LetX beafibreofZ. Then
N(X −Xalg,T) ≤ c(Z,(cid:3))T(cid:1).
TheexampleX = {(cid:3)x,y,z(cid:4) ∈ R3 : z = xy,x,y ∈ [2,3]},forwhichXalg = {(cid:3)x,y,z(cid:4) ∈
X : y ∈ Q},showsthatXalgisnot,ingeneral,semialgebraic(orevendefinable: adefinableset
hasonlyfinitelymanyconnectedcomponents). Nevertheless,itmightbesupposedthat,forany
X and(cid:3),thereisasemialgebraicsetX ⊂ X andaconstantc(X,(cid:3))suchthatN(X−X ,T) ≤
(cid:1) (cid:1)
c(X,(cid:3))T(cid:1). This is not the case: Consider X = {(cid:3)x,y(cid:4) : 0 < x < 1, 0 < y < ex}. Then
Xalg = X but X is not semialgebraic (otherwise its bounding graph y = ex,x ∈ [0,1] would
be semialgebraic). So N(X −X ,T) >> T4 for any semialgebraic X ⊂ X. However, it is
(cid:1) (cid:1)
possibletofindadefinableX ⊂ Xalg withthedesiredproperty;indeed,foradefinablefamily
(cid:1)
Z thesetsX maybetakentobefibresofadefinablefamilyW(Z,(cid:3)) ⊂ Z,andthisisthefinal
(cid:1)
versionoftheresulttobeproved.
1.10. Theorem. (Final version) Let Z ⊂ Rn ×Rm be a definable family and (cid:3) > 0. There is
a definable family W = W(Z,(cid:3)) ⊂ Z and a constant c(Z,(cid:3)) with the following property. Let
y ∈ Y. PutX = X andX = X . ThenX ⊂ Xalg and
Z,y (cid:1) W,y (cid:1)
N(X −X ,T) ≤ c(Z,(cid:3))T(cid:1).
(cid:1)
Note that this version makes a nontrivial assertion in situations, like the example above,
inwhich(X −Xalg)(Q)isemptybutXalg isnotdefinable.
The diophantine part of the proof follows the strategy going back to [4]. The heart of the
analyticpartoftheproofisthepossibilityofacertainuniformparameterizationofthefibresX
inadefinablefamily. TheuniformityrequiredisinthenumberofC(r)maps(0,1)dim(X) → X
requiredtocoverX,andatthesametimeinboundsonthesizesofalltheirpartialderivativesup
tosomeprescribedfiniteorderr. Thisisachievedin§2–5,byestablishingano-minimalversion
5
of Gromov’s Algebraic Reparameterization Lemma (see [11, page 232]; itself a refinement of
a method of Yomdin [36, 37]) for obtaining such parameterizations of closed semialgebraic
sets.) Our o-minimal version of Gromov’s Reparameterization Lemma may well find other
applications.
(Noteadded: Gromov’sproofisverybrief;acarefulproofhasbeengivenbyBurguet[5].)
In [22] a conjecture is also made about integer points on the dilation of a compact sub-
analytic set. That conjecture is essentially (though not strictly) weaker than the corresponding
statement about rational points, and is also affirmed here, for (bounded) definable sets, in §8.
Otherresultsonintegerpointsondefinablecurvesareobtainedin[35].
While, as indicated, the estimate N(X −Xalg,T) = O (T(cid:1)) cannot be improved for
X,(cid:1)
globally subanalytic sets, a much better estimate might be anticipated for other o-minimal
structureswherewehavemorecontroloverthedefinablesets. Forexample:
1.11. Conjecture. Let X be definable in R . Then there are constants c (X),c (X) such
exp 1 2
that(forT ≥ e)
N(X −Xalg,T) ≤ c (X)(logT)c2(X).
1
It should be observed that the results here give, even in the original situation of one
dimensionalsetsX : y = f(x)inR2,anextensionofthepreviousresults: Curvesdefinablein
o-minimal structures may have derivatives that degenerate at an endpoint, or they may be not
analytic at any point (e.g. the examples in [26]). The previous results may be inapplicable in
such situations, and thus the present results may well find further application in the circle of
problems[27,19,28]thatprovidedtheiroriginalmotivation.
In this paper, A ⊂ B means that A is a subset of (possibly equal to) B. The cardi-
nality of a set A is denoted #A, and N denotes the set of nonnegative integers. The letters
i,j,k,(cid:1),m,n,r,darereservedexclusivelytorangeoverN.
Acknowledgements
Theauthorsareverygratefultotherefereesfortheirmosthelpfulcommentsandsugges-
tions. The first author thanks A. Venkatesh and L. van den Dries for assistance. He is grateful
to the Mathematical Institute, Oxford, and in particular A. Lauder and R. Heath-Brown, for
hospitality during a visit in which parts of this paper were written. His work was partly sup-
portedbyMcGillUniversityandagrantfromNSERC.Thesecondauthorisgratefultoallthe
staff at, and visitors to, the Isaac Newton Institute for Mathematical Sciences for making the
Programme on Model Theory and Applications to Algebra and Analysis (January 17 –July 15
2005) so successful. The contents of this paper were first presented there, and he thanks, in
particular,ZoeChatzidakisandSergeiStarchenkoforextremelyhelpfulcomments.
2. Reparameterization(afterYomdin-Gromov)
For §2–5 we fix an o-minimal structure S over a real closed field M; definable will mean
definableinS (see[7]). AlthoughweareultimatelyonlyinterestedinR,thegreatergenerality
actuallysimplifiestheargumentsherebecauseitguaranteesacertain“uniformityinparameters”
thatwouldbeabsentifwerestrictedourattention tostructuresoverR.
6
Recallthatanelementa ∈ M iscalledfiniteif|a| ≤ cforsomec ∈ N(weassumethatQ
is identified with the prime subfield of M). A finite element of M will also be called strongly
bounded. An n-tuple of elements of M is strongly bounded if all its coordinates are, and a
definablesubsetofMn isstronglyboundedifthereisafixedfiniteboundforallthecoordinates
ofallitselements. Further,adefinablefunctionisstronglyboundedifitsgraphis(equivalently,
ifitsdomainandrangeare).
2.1. Definition. Let X ⊂ Mn be definable. A definable function φ : (0,1)(cid:3) → X, where
(cid:1) = dimX,iscalledapartialparameterizationofX. AfinitesetS ofpartialparametrizations
of X is called a parameterization of X if ∪φ∈Srange(φ) = X. (Of course standard notation
like“(0,1)”referstoitsnaturalinterpretation inM.)
We shall be interested in various extra conditions on the functions in such an S. In
particular, it is not hard to show, using the C(r)-cell decomposition theorem ([7]), that every
boundedsethasaC(r)-parameterization. Weshallbeinterestedinboundingthederivatives.
2.2. Definition. AparameterizationS(ofsomedefinablesetX)iscalledanr-parameterization
if every φ ∈ S is of class C(r) and has the property that φ(α) is strongly bounded for each
α ∈ NdimX with|α| ≤ r,where|α|isthesumofthecoordinatesofα.
2.3. Theorem. (Reparametrization Theorem [after Gromov]) For any r ∈ N and any strongly
bounded,definablesetX,thereexistsanr-parameterizationofX.
Thereisalsoaversionforfunctions.
2.4. Definition. Suppose that S is an r-parameterization of the definable set X ⊂ Mm and
thatF : X → Mn isadefinablefunction. ThenwesaythatS isanr-reparameterizationofF
if,foreachφ ∈ S,F ◦φisofclassC(r) and(F ◦φ)(α) isstronglyboundedforallα ∈ NdimX
with|α| ≤ r.
2.5. Theorem. Foranyr ∈ Nandanystronglybounded,definablefunctionF,thereexistsan
r-reparameterizationofF.
Thenext3sectionsaredevotedtotheproofoftheorems2.3and2.5. Itwillbeconvenient
to assume that M is ℵ -saturated, in particular non-archimedean, in these sections. This can
0
alwaysbeassumedbytakingasuitableelementaryextensionandnotingthatthestatementsof
2.3and2.5pullbacktotheoriginalstructure.
3. Theunaryfunctioncase
Thereisaverysimple,butcrucial,analytictrickattheheartoftheproofof2.5whichwe
nowstateandprove. Indeed,therestoftheargumentisjustacaseoforganizingtheinduction
carefully.
3.1. Lemma. Let r ≥ 2 and suppose that f : (0,1) → M is a definable function of class
C(r) with f(j) strongly bounded for 0 ≤ j ≤ r − 1. Suppose further that |f(r)| is (weakly)
decreasing. Defineg : (0,1) → M by
g(x) = f(x2).
7
Theng(j) isstronglyboundedfor0 ≤ j ≤ r.
(cid:3)
Proof. By the chain rule (applied in M), g(i)(x) = i ρ (x).f(j)(x2), for each i =
j=0 i,j
0,1,...,r and x ∈ (0,1), where each ρ is a polynomial with integer coefficients (of degree
i,j
j,infact).
Now, by our hypothesis on f, all summands are strongly bounded except, possibly, the
one with i = j = r. One easily checks that this summand is2rxrf(r)(x2). Let c be a positive
integer strongly bounding the function f(r−1) and suppose, for a contradiction, that there is a
somex ∈ (0,1)with|f(r)(x )| > 4c/x . BytheMeanValueTheorem(appliedinM),there
0 0 0
is some ξ ∈ [x /2,x ] such that f(r−1)(x )−f(r−1)(x /2) = f(r)(ξ).(x −x /2). But by
0 0 0 0 0 0
ourhypothesisonf(r) wehave|f(r)(ξ)| ≥ |f(r)(x )| > 4c/x . Hence
0 0
4c
2c ≥ |f(r−1)(x )−f(r−1)(x /2)| > (x −x /2) = 2c.
0 0 0 0
x
0
Thiscontradiction showsthat
4c
|2rxrf(r)(x2)| ≤ 2rxr
x2
for all x ∈ (0,1), and the right-hand side here is bounded by 2r+2c since r ≥ 2. Thus g(i) is
stronglyboundedfori = 0,1,...,r,andthelemmaisproved.
3.2. Lemma. Let F : (0,1) → M be a definable, strongly bounded function. Then F has
a 1-reparameterization, S say, with the additional property that, for each φ ∈ S, either φ or
F ◦φisapolynomial(restrictedto(0,1))withstronglyboundedcoefficients.
Proof. By o-minimality, choose elements a = 0 < a < ... < a < a = 1 of M so that,
0 1 p p+1
for each i = 0,1,...,p, F is of class C(1) and satisfies either |F(cid:3)| ≤ 1 throughout (a ,a )
i i+1
or|F(cid:3)| ≥ 1throughout(a ,a ).
i i+1
Inthefirstcase,defineφ : (0,1) → M byx (cid:13)→ (a −a )x+a .
i i+1 i i
In the second case (when F is certainly strictly monotone and continuous on (a ,a ))
i i+1
we set b = lim F(x),b = lim F(x) and define φ : (0,1) → M by x (cid:13)→
i x→a+ i+1 x→a− i
i i+1
F−1((b −b )x+b ).
i+1 i i
Ineithercase,range(φ ) = (a ,a )andbothφ andF ◦φ areofclassC(1) throughout
i i i+1 i i
(0,1) with strongly bounded derivatives. Further, at least one of these functions is linear
with coefficients in [−1,1]. It is now clear that S = {φ ,φ ,...,φ ,aˆ,...,aˆ} is a 1-
0 1 p 1 p
reparameterizationofF withtherequiredadditionalproperty,whereeachaˆ denotestheconstant
i
functionon(0,1)withvaluea .
i
3.3. Lemma. Let r ≥ 1 and suppose that F : (0,1) → M is a definable, strongly bounded
function. ThenF hasanr-reparameterization(withtheadditionalpropertythat,foreachφin
it,eitherφorF ◦φisapolynomial(restrictedto(0,1))withstronglybounded coefficients).
Proof. Theproof(ofthewholestatement,includingtheparentheticalproperty)isbyinduction
on r. The case r = 1 being Lemma 3.2, suppose that r ≥ 2 and that S is an (r − 1)-
reparameterizationofF withtheadditionalproperty. Letφ ∈ S andwrite{φ,F ◦φ} = {g,h}
8
where g is a polynomial (restricted to (0,1)) with strongly bounded coefficients. Thus, in
particular,g(i) existsandisstronglyboundedforalli. However,weonlyknowthath(i) exists,
is continuous, and is strongly bounded for i = 0,...,r −1. In order to apply Lemma 3.1 we
use o-minimality to pick elements 0 = a < a < ... < a < a = 1 in M (depending
0 1 p p +1
φ φ
on φ) so that, for each i = 0,...,p , the function h is of class C(r) on (a ,a ) and |h(r)| is
φ i i+1
(weakly)monotonicon(a ,a ).
i i+1
Letθ : (0,1) → (0,1)bedefinedby
φ,i
(cid:4)
(a −a )x+a , if |h(r)| is (weakly) decreasing,
θ (x) = i+1 i i
φ,i (a −a )x+a , if |h(r)| is (weakly) increasing.
i i+1 i+1
(Wechoosethefirstoption,say,if|h(r)|isconstant.)
It is immediate from the inductive hypothesis that h◦θ : (0,1) → M is of class C(r),
φ,i
andthat(h◦θ )(i)isstronglyboundedfori = 0,...,r−1. Further,|(h◦θ )(r)|is(weakly)
φ,i φ,i
decreasing. Letρ : (0,1) → (0,1)betheC(∞) bijectionsendingxtox2. ThenbyLemma3.1,
the function h◦θ ◦ρ : (0,1) → M has strongly bounded i-th derivative for i = 0,...,r.
φ,i
Ofcourse,thefunctiong◦θ ◦ρisstillapolynomialwithstronglyboundedcoefficientsand
φ,i
{h◦θ ◦ρ,g◦θ ◦ρ} = {φ◦θ ◦ρ,F ◦(φ◦θ ◦ρ)}. Noticealsothatasivariesfrom0
φ,i φ,i φ,i φ,i
top ,range(φ◦θ ◦ρ)coversrange(φ)apartfromfinitelymanypoints. Soweonlyhaveto
φ φ,i
addfinitelymanyconstantfunctions(takingvaluesin(0,1))totheset{φ◦θ ◦ρ : φ ∈ S}in
φ,i
orderforittobecomeanr-reparameterizationofF withtherequiredadditionalproperty. This
completestheinductionandtheproofofthelemma.
3.4. Corollary. LetX beastronglyboundedsubsetofM andF : X → M astronglybounded
function. Thenforallr ≥ 1,F hasanr-reparameterization.
Proof. Since X is a (finite) union of strongly bounded intervals and points, it clearly has
an r-parameterization, S say, by linear and constant functions. Now use Lemma 3.3 to r-
reparameterize each funcion F ◦ φ : (0,1) → M, for φ ∈ S, and take the union of these
reparameterizations.
We now proceed to the case of functions taking values in Mn for n ≥ 2. However, there
isnothingspecialaboutunaryfunctionsinthisprocess,sowedothegeneralcasenow.
3.5. Lemma. Letm,r ≥ 1andassumethateverydefinable,stronglyboundedfunctionwithdo-
mainasubsetXofM(cid:3)(forsome(cid:1) ≤ m)andrangeasubsetofM,hasanr-reparameterization.
Then for any n ≥ 1, the same is true for such functions having range a subset of Mn (and
domainX).
Proof. It is clearly sufficient (by the obvious inductive argument) to show that if n ≥ 2 and
F : X → Mn−1,f : X → M are definable, strongly bounded functions such that F has
an r-reparameterization, then so does the function (cid:3)F,f(cid:4) : X → Mn, where we may as well
supposethatX isadefinable(stronglybounded)subsetofMm.
So let S be an r-reparameterization of F, and let φ ∈ S. Say φ : (0,1)(cid:3) → X where
(cid:1) = dim(X) ≤ m. Applythehypothesisofthelemmatothefunctionf ◦φ : (0,1)(cid:3) → M,to
obtainanr-reparameterizationofit,T say. Theneachψ ∈ T hasdomain(0,1)(cid:3),anditclearly
φ φ
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