Table Of Content1 Introduction and Content
1.1 Motivations
This book deals with several related topics:
– Geometryofrealsemialgebraicandtamesets(i.e.setsdefinableinsomeo-
minimalstructureon(R,+,.)),withthestresson“metric”characteristics:
lengths, volumes in different dimensions, curvatures etc...
– Behaviour of these characteristics under polynomial mappings.
– Integral geometry, especially the so-called Vitushkin variations, with the
stress on applications to semialgebraic and tame sets.
– Geometry of critical and near critical values of differentiable mappings.
Some fractal geometry naturally arising in this context.
Below we give a short description of each of these topics, their mutual
dependance and logical order. Motivation for the type of question asked in
this book, comes from several different sources: the main ones are Differen-
tial Topology, Singularity Theory, Smooth Dynamics, Control, Robotics and
Numerical Analysis.
One of the main analytic results, underlying most basic constructions of
Differential Analysis, Differential Topology, Differential Geometry, Differen-
tial Dynamics, Singularity Theory, as well as nonlinear Numerical Analysis,
is the so-called Sard (or Morse-Sard, or Morse-Morse-Sard, see [Morse 1,2],
[Mors], [Sar 1-3]) theorem. It asserts that the set of critical values of a suffi-
cientlysmoothmappinghasmeasurezero.Mostlythistheoremappearsasan
assumption,thatthesetY(c)ofsolutionsofanequationf(x)=cisasmooth
submanifoldofthedomainofx,foralmostanyvaluecoftherighthandside
(in the semialgebraic or tame case, an asymptotic version of the Morse-Sard
theorem(see[Rab],[Kur-Orr-Sim])showsthatf inducesafibrationoverthe
connected components of the “good values” c).
Another typical appearence of the Morse-Sard theorem is in the form of
varioustransversalitystatements:byasmallperturbationofthedatawecan
always achieve a situation where all the submanifolds of interest intersect
one another in a transversal way. Fig.1.1 shows a non-transversal (a) and
transversal (b, c) intersections of two plane curves.
Y.YomdinandG.Comte:LNM1834,pp.1–22,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Introduction and Content
Fig. 1.1.
Technically, the Morse-Sard theorem is a rather subtle fact. It is well
known since the classical examples of Whitney [Whi 1], that the geometry of
the critical points and of the critical values of a differentiable function can
be very complicated. In particular, the requirement that the function must
haveatleastthesamenumberofderivativesasthedimensionofthedomain,
cannotberelaxed.Andusuallyanalyticfactsthatincorporateinanessential
way the existence and the properties of the high order derivatives, are deep,
both conceptually and technically.
However, in classical applications the Morse-Sard theorem appears as a
background fact, as a default assumption, that all the transversality and
regularity properties required can be achieved by small perturbations of the
data.Theinterplaybetweenthehighorderanalyticstructureofthemappings
involved and their geometry rarely becomes apparent. The main reason is
that the classical Morse-Sard theorem is basically qualitative. Its conclusion
appears as an “existential” fact and it provides no quantitative information
on the solutions, submanifolds etc... in question.
Averynaturalquantitativesettingoftheproblems,coveredbytheMorse-
Sard theorem, is possible. In the case of transversality, the question is: given
a maximal size of perturbations allowed, how strong a transversality of the
perturbed submanifolds can be achieved? (For plane curves the transversality
can be measured just by the angle between the curves at their intersection
points.Fig.1.1cpresentsastrongtransversality,incontrasttoaweakonein
Fig.1.1b).
Concerning the set Y(c) of solutions of the equation f(x) = c, a natural
quantitative question is: How much of the complexity of Y(c) (geometric or
topological)canbeeliminatedbyaperturbationofcwithintheallowedrange?
What is the average complexity (with respect to c) of Y(c)?
In nonlinear Numerical Analysis, a typical conclusion, provided by the
classical Morse-Sard theorem, is that with probability one certain determi-
nants do not vanish. However, to organize computations in a stable way it is
necessary to get a quantitative information: how big are these determinants
usually ?
The importance of this sort of quantitative information has been realized
during the last decades in many fields. In the study of the complexity of
1.1 Motivations 3
algorithms (especially, in the work of Shub and Smale on complexity of the
Newton type algorithms [Shu-Sma 1-7], [Sma 3,4]), quantitative considera-
tions of the above type appear as one of the main tools (although mainly in
situations where a direct treatment without Morse-Sard’s theorem is possi-
ble).
In a recent study of high order numerical algorithms ([Eli-Yom 1-5], [Bri-
Yom6],[Bri-Eli-Yom],[Bic-Yom],[Wie-Yom],[Yom24],[Y-E-B-S])itbecame
apparent that Quantitative Morse-Sard theorem and quantitative transver-
sality may be crucially important in efficient organization of a high order
data and in its efficient processing. We discuss this issue in some detail in
Section 1.1.3, Chapter 1, and in Section 10.1.4, Chapter 10 below.
InDifferentialDynamics,anumberof“quantitative”problemshavebeen
posed by M. Gromov in the early eighties.
These concerned a quantitative behavior of periodic points, estimates for
thevolumegrowthandentropyetc...(See[Gro1-4]).A“QuantitativeKupka-
Smale theorem”, bounding a typical quantitative behavior of periodic points
and conjectured by M. Gromov, has been obtained in [Yom 4]. Very recently
striking results in this direction have been obtained by Kaloshin [Kal 1-
4] (some of these dynamical results are briefly discussed in Section 10.1.3,
Chapter 10 below).
Importantapplicationsofquantitativetransversalityinsymplecticgeome-
try appeared recently in S. K. Donaldson’s papers ([Don 1-3], see also [Sik]).
These results have been further extended in [Aur], [Ibo] and other publica-
tions.
As for the Morse-Sard theorem itself, its sharpest quantitative version,
concerning entropy dimension, has been obtained in [Yom 1]. Further appli-
cations, answering, in particular, a part of the quantitative questions above,
appeared in [Yom 3,4,7,10,17,18,20]. More recently additional geometric and
analytic information, related to different versions of the Morse-Sard theo-
rem,concerningHausdorffmeasureanddimension,hasbeenobtainedin[Bat
1-6], [Bat-Mor], [Bat-Nor], [Com 1], [Nor 1-4], [Nor-Pug], [Roh 1-3], [Yom
13-15,19], culminating in [Mor], in which the sharpest possible statement is
given.Concerningsingularvaluesatinfinityandtheso-calledMalgrangecon-
dition, one can see [Kur-Orr-Sim] for the semialgebraic case and [D’Ac] for
the o-minimal case.
Oneofthemaingoalsofthisbookistogiveaproofandan“explanation”
of the quantitative Morse-Sard theorem and related results. This is done via
the study of the same questions first for polynomial (or tame) mappings. In-
deed, while the classical Morse-Sard theorem is trivial for polynomials (criti-
calvaluesalwaysformasemialgebraicsetofadimensionsmallerthanthatof
the ambient space, and thus have Lebesgue measure zero), the quantitative
questions above turn out to be nontrivial and highly productive. They are
answeredinthisbookbyacombinationofthemethodsofRealSemialgebraic
and Tame Geometry and Integral Geometry.
4 1 Introduction and Content
One of the important advantages of this approach is that it allows one to
separate the role of high differentiability and that of algebraic geometry in
a smooth setting: all the geometrically relevant phenomena appear already
forpolynomialmappings.Thegeometricpropertiesobtainedare“stablewith
respect to approximation”, and so can be imposed on smooth functions via
polynomialapproximation.Theonlyroleofhighdifferentiabilityistocontrol
the rate of this approximation. (In fact, the high order differentiability turns
out to be not relevant at all in this circle of problems ! It is the rate of
approximationbysemialgebraicfunctions,thatreallycounts.SeeSection10.2
below).
Now the study of metric Semialgebraic Geometry with the above appli-
cations in view, essentially forces us to extend the tools beyond the usual
lengths, areas etc... It is explained in detail below why using metric entropy
(theminimalnumberofballsofaprescribedradius,coveringagivenset)and
multidimensional variations (the average number of connected components
in plane crossections of different dimensions) is most natural and rewarding
in our setting.
In conclusion, let us express our hope that the results and methods pre-
sented in this book form only a beginning of the future “Quantitative Singu-
larity Theory”. The ultimate need for this theory is by now realized in many
fields of mathematics. Quantitative Sard theorem, Quantitative Transversal-
ity and “Near Thom-Boardman Singularities” treated in this book definitely
belong to this future theory, whose possible contours are discussed in some
detail in Section 10.3.7 below.
1.2 Content and Organization of the Book
In the next section of this introduction we explain the main ideas of the
semialgebraic part of the book, using a rather instructive example of the
motion control problem in robotics. In the last section of the introduction
we give an accurate proof of the simplest version of the generalized Sard
theorem.Thisproofillustratesinasimpleandtransparentform(andwithout
technicalities, unavoidable in a general setting) a good part of the ideas and
methods developed below.
Chapter2isdevotedtoapreciseintroductionandaratherdetailedstudy
of the metric entropy of subsets of Euclidean spaces. We believe that the
“transversality” results of Proposition 2.2 and Corollary 2.3, as well as a
geometric interpretation of the entropy dimension, given by Theorem 2.9,
are new.
In Chapter 3 we recall the theory of multidimensional variations, develo-
ped by A. G. Vitushkin ([Vit 1,2]), L.D. Ivanov ([Iva 1,2]), and others.
1.2 Content and Organization of the Book 5
Ingeneral,tohandlemultidimensionalvariationsisnotaneasytask.Most
of the results for which this theory was initially developed (in particular,
restrictionsoncompositionrepresentabilityofsmoothfunctions[Vit3]),had
been later obtained by different (easier) methods. As a result, today it is not
easy to find a presentation of this theory, especially in English. We believe
that multi-dimensional variations, as applied to semialgebraic sets, give a
very convenient and adequate geometric tool. Indeed, by definition, the i-
th variation of A ⊆ Rn, V (A), is the average of the number of connected
i
componentsofthesectionA∩P overallthe(n−i)-dimensionalaffineplanes
P inRn.ForA-semialgebraic,thenumberofconnectedcomponentsofA∩P
is always bounded in terms of the diagram of A (i.e. of the degrees of the
defining polynomials and of their set-theoretic formula), and hence to bound
variations we need just to estimate the size of various projections of A.
Ontheotherhand,thefollowingbasicinequalityrelatesmultidimensional
variations with metric entropy: For any A⊆Rn,
(cid:1)n
1
M((cid:1),A) (cid:1)C(n) V (A)( )i .
i (cid:1)
i=1
We give in Section 1.3 a rather detailed introduction to the theory of varia-
tions, in particular providing a complete proof of the above inequality for a
general subset A⊆Rn (following [Zer]). We hope that this section, together
with Section 5, where variations of semialgebraic sets are studied, can fill to
some extent the gap in the literature on this subject.
In Chapter 4 we give some generalities on semialgebraic and tame sets,
andproveexplicitly(andwithexplicitbounds)thepropertiesrequiredinthe
rest of the book: bounds on the number of connected components, “covering
theorems” (such as Theorem 1.3 stated below), etc...
Chapter 5 is devoted to variations of semialgebraic and tame sets. We
stress the properties which are not true in general: comparison of variations
oftwotamesets,closetooneanotherintheHausdorffmetric,inparticular,of
asetanditsδ-neighborhood,correlationsbetweenvariationsofthesameset,
in different dimensions (in general, V (A) for different i are “independent”),
i
bounds on the radius of a maximal ball, contained in a δ-neighborhood of a
set, etc...
Chapter 6 has a somewhat technical character. To study the behavior
of tame and semialgebraic sets under mappings (in the same category), we
have to measure properly the size of the first differential of the mappings.
Roughly, we use as the “sizes” of a linear mapping (in different dimensions)
the semiaxes of the ellipsoid, which is the image of the unit ball under this
mapping. This leads to some exterior algebra (sometimes not completely
trivial, especially as we want to deal with plane sections and integration).
6 1 Introduction and Content
Chapter 7 contains the main results of this book, as far as the tame
(semialgebraic) sets and mappings are concerned. Basically they have the
following form: assuming that the size of the differential Df of f is bounded
(inonesenseoranother)onasetA,weestimatevariations(andhencemetric
entropy)oftheimagef(A)(Theorems7.1and7.2).Wededucefromtheresult
the quantitative Morse-Sard theorem in the polynomial case (Theorem 7.5).
In particular, we obtain, as a special case, Theorem 1.6 below.
Chapter8continuesthelineofChapter7,withsomewhatmorespecialre-
sults,relatedto“quantitativetransversality”ononeside,andtothebehavior
of mappings on more complicated singularities.
Finally, in Chapter 9 we apply the results of Chapters 7 and 8 to map-
pings of finite smoothness. The main tool is a Taylor approximation of the
mappings; then we use appropriate “semialgebraic” results. Since these re-
sults“surviveunderapproximation”,itremainstocountthetotalnumberof
Taylor polynomials in the approximation. Consequently, the results have the
formofthecorresponding“semialgebraic”estimatewitha“remainderterm”,
taking into account a finite smoothness (Quantitative Morse-Sard theorem,
Theorem 9.2). Considered from the point of view of Differential Analysis
and Topology, the results of Chapter 9 give far-reaching improvements and
generalizations of the usual Morse-Sard theorem.
In Chapter 10 we give a short overview of some additional applications
of the results and methods presented in this book, and of some directions of
their further development. The applications include:
– Maxima of smooth families
– Further applications in differential topology
– Smooth Dynamics
– Numerical Analysis
In some details the Semialgebraic Complexity of functions is defined and
discussed.
We discuss briefly the following directions of further development:
– Asymptotic critical values
– Morse-Sard theorem in Sobolev spaces
– Real equisingularity
– “Ck-resolution” of semialgebraic sets and mappings
– Bernstein type inequalities for algebraic functions
– Polynomial Control problems
– Quantitative Singularity Theory
1.3 The Motion Planing Problem in Robotics as an Example 7
1.3 The Motion Planing Problem
in Robotics as an Example
Probably the most natural example, where many of the results of this book
have immediate and direct interpretation, is provided by various aspects of
the so-called “Motion Planning Problem” in Robotics. This example allows
onetounderstandthepowerofthemethodsdiscussedinthisbook,aswellas
theirlimitations.Moreover,weshalltrytoshowinthisexamplewhatshould
bedoneingeneralinordertotransformtheenormousanalyticpowerofhigh
order analytic and geometric methods into efficient computational tools.
The problem of motion planning is real, important and difficult, and it
may be analyzed and (in principle) solved completely in the framework of
Semialgebraic Geometry (although, as we explain below, to deliver its full
power, Semialgebraic geometry must be combined with Singularity Theory
and with a clever data representation). The main objects of this book, like
“effectivecurvesselection”insidesemialgebraicsets,coveringofsemialgebraic
sets via polynomial mappings, critical and near-critical points and values
of polynomial mappings, – become directly visible in motion planning. The
equationsarisinginthesimplestexamplesareofreasonabledegrees,andthey
canbeexplicitelysolvedandanalyzedonpopularsymbolicalgebrapackages.
On the other hand, such practical experiments show immediately the (very
narrow) limits of a direct applicability of algebro-geometric methods.
Allthisjustifies,inourview,aratherdetailedpresentationofthemotion
planning problem, given below. This presentation follows mostly [Sch-Sha],
[Eli-Yom 3], [Tan-Yom] and [Sham-Yom].
LetB beasystemcomprisingacollectionofrigidsubparts,someofwhich
might be attached to each other at certain joints, while others might move
independently. Suppose B has a total of (cid:3) degrees of freedom, that is, each
placementofB canbespecifiedby(cid:3)realparameters,eachrepresentingsome
relationship(orientation,displacement,etc...)betweencertainsubpartsofB.
Suppose further that B is free to move in a two- or three-dimensional space
amidst a collection of obstacles O whose geometry is known. Typical values
of (cid:3) range from 2 (for a rigid object translating on a planar floor without
rotating) to 6 (the typical number of joints for a manipulator arm). The
values can also be much larger – for example, when we need to coordinate
the motion of several independent systems in the same workspace.
Let P ⊆R(cid:1) denote the space of the parameters of our problem.
The motion-planning problem for B is: given an initial placement Z1
and a desired target placement Z2 of B, determine whether there exists a
continuous obstacle-avoiding motion of B from Z1 to Z2, and, if so, plan
such a motion.
Let us consider two examples. The first one is shown in Fig.1.2. This is
a plane “robotic manipulator”, consisting of two bars b1 and b2. The bar b1
8 1 Introduction and Content
has its endpoint e1 fixed at the origin, and the endpoint of b2 is fixed at the
second endpoint e2 of b1. Both b1 and b2 can rotate freely at e1 and e2.
O1, O2 and O3 denote the obstacles, and the initial placement Z1 and
the desired target placement Z2 are shown on the picture. Taking as free
parameters the angles ϕ1 and ϕ2 shown in Fig.1.2, we get the space P of
parameters as the square [0,2π]×[0,2π] in R2 (or rather a torus T2 – this
more accurate topological representation sometimes helps).
Fig. 1.2.
Anotherexampleofamotion-planningproblemisrepresentedinFig.1.3.
We have to move the plane rectangle B from the initial position Z1 into
the target position Z2 avoiding the obstacles O1,...,Oz. (One can consider
thistaskasaversionofawell-knowngeometricproblem:whatistheminimal
possible area of a plane domain, inside which we can turn a length 1 needle
180 degrees (see [Tao]))?
Fig. 1.3.
1.3 The Motion Planing Problem in Robotics as an Example 9
Here we have 3 degrees of freedom; as the parameters can be taken to
be the coordinates (x,y) of the barycenter b of B and the rotation angle ϕ.
Probably a direct examination of these problems will not provide a definite
answer (at least for most readers). However, the solution will be greatly
simplifiedifwepasstotheso-called“freeconfigurationspace”oftheproblem.
Generally the free configuration space of the moving system B denoted FP
is the (cid:3)-dimensional parametric space of all free placements of B (the set of
placementsofB inwhichB doesnotintersectanyobstacle).Eachpointz in
FP is a (cid:3)-tuple giving the values of the parameters controlling the (cid:3) degrees
of freedom of B at the corresponding placement. Clearly, finding a motion
from a placement Z1 represented by Z1 ∈ P, to Z2 represented by Z2, is
equivalent to joining Z1 and Z2 by a continuous path in FP.
The free configuration space FP of the first problem is shown in Fig.1.4,
together with the initial and target configurations Z1,Z2.
Fig. 1.4.
Now one sees immediately that the solution exists, since Z1 and Z2 be-
long to the same connected component of FP. Three of the “control (or
configuration) trajectories” joining Z1 and Z2 are shown in Fig.1.4, and the
corresponding evolution of the manipulator is given in Fig.1.5.
This figure shows one of the three solutions, represented on Fig.1.4,
namely ρ1. It consists of 4 rotations (3 of them are consecutive, illustrated
by arcs 1, 1(cid:1), 2 and 3).
Thus the main difficulty in solving the motion planning problem consists
in the construction of the free configuration space. This construction is non-
trivialalreadyinthefirstexampleconsidered.Inthesecondexamplethefree
configuration space FP is fairly complicated: it looks like a spiralled worm-
10 1 Introduction and Content
Fig. 1.5.
holeinthethree-dimensionalcube,andwedonotshowithere.However,the
solution turns out to exist, and is shown in Fig.1.6.
Fig. 1.6.
Now the basic fact is that if each part of the system B and each obsta-
cle O are semialgebraic (i.e., representable by a finite number of polynomial
equations, inequalities and set-theoretic operations), then the free configura-
tionspace FP is semialgebraic,andcanbecomputedeffectivelyfromB and
O.
Thereexistsalsoaneffectiveproceduretodecidewhethertwogivenpoints
belong to the same connected component of a given semialgebraic set. Con-
sequently, for semialgebraic data (which is a very natural assumption) the
motion planning problem can be effectively solved. See [Sch-Sha] for details.
Important remark. “Effectively” does not mean “efficiently”! The com-
plexity of the algorithms, based on the direct approach as above and using
