30 research outputs found
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
We prove new upper bounds on homotopy and homology groups of o-minimal sets
in terms of their approximations by compact o-minimal sets. In particular, we
improve the known upper bounds on Betti numbers of semialgebraic sets defined
by quantifier-free formulae, and obtain for the first time a singly exponential
bound on Betti numbers of sub-Pfaffian sets.Comment: 20 pages, 2 figure
Topology of definable Hausdorff limits
Let be a set definable in an o-minimal expansion of the
real field, be its projection, and assume that the non-empty
fibers are compact for all and uniformly bounded,
{\em i.e.} all fibers are contained in a ball of fixed radius If
is the Hausdorff limit of a sequence of fibers we give an
upper-bound for the Betti numbers in terms of definable sets
explicitly constructed from a fiber In particular, this allows to
establish effective complexity bounds in the semialgebraic case and in the
Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative
closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian
functions in a way that is adapted to complexity problems. Our results can be
used to estimate the Betti numbers of a relative closure in the
special case where is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and
notations in an attempt to be clearer, references adde
Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time
In this paper we describe an algorithm that takes as input a description of a
semi-algebraic set , defined by a Boolean formula with atoms of
the form for
and outputs the first Betti numbers of ,
The complexity of the algorithm is where where s =
#({\mathcal P}) and which is
singly exponential in for any fixed constant. Previously, singly
exponential time algorithms were known only for computing the Euler-Poincar\'e
characteristic, the zero-th and the first Betti numbers
Topological complexity of the relative closure of a semi-Pfaffian couple
Gabrielov introduced the notion of relative closure of a Pfaffian couple as
an alternative construction of the o-minimal structure generated by
Khovanskii's Pfaffian functions. In this paper, use the notion of format (or
complexity) of a Pfaffian couple to derive explicit upper-bounds for the
homology of its relative closure.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti
numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve
On the number of homotopy types of fibres of a definable map
In this paper we prove a single exponential upper bound on the number of
possible homotopy types of the fibres of a Pfaffian map, in terms of the format
of its graph. In particular we show that if a semi-algebraic set , where is a real closed field, is defined by a Boolean formula
with polynomials of degrees less than , and
is the projection on a subspace, then the number of different homotopy types of
fibres of does not exceed . As applications
of our main results we prove single exponential bounds on the number of
homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials
with bounded additive complexity. We also prove single exponential upper bounds
on the radii of balls guaranteeing local contractibility for semi-algebraic
sets defined by polynomials with integer coefficients.Comment: Improved combinatorial complexit
Bounding Betti Numbers of Sets Definable in O-Minimal Structures Over the Reals
A bound for Betti numbers of sets definable in o-minimal structures is
presented.
An axiomatic complexity measure is defined, allowing various concrete
complexity measures for definable functions to be covered. This includes common
concrete measures such as the degree of polynomials, and complexity of Pfaffian
functions.
A generalisation of the Thom-Milnor Bound for sets defined by the conjunction
of equations and non-strict inequalities is presented, in the new context of
sets definable in o-minimal structures using the axiomatic complexity measure.
Next bounds are produced for sets defined by Boolean combinations of equations
and inequalities, through firstly considering sets defined by sign conditions,
then using this to produce results for closed sets, and then making use of a
construction to approximate any set defined by a Boolean combination of
equations and inequalities by a closed set.
Lastly, existing results for sets defined using quantifiers on an open or
closed set are generalised, using a construction from Gabrielov and Vorobjov to
approximate any set by a compact set. This results in a method to find a
general bound for any set definable in an o-minimal structure in terms of the
axiomatic complexity measure. As a consequence for the first time an upper
bound for sub-Pfaffian sets defined by arbitrary formulae with quantifiers is
given. This bound is singly exponential if the number of quantifier
alternations is fixed.Comment: 82 page, PhD thesi
Erdos-Szekeres-type statements: Ramsey function and decidability in dimension 1
A classical and widely used lemma of Erdos and Szekeres asserts that for
every n there exists N such that every N-term sequence a of real numbers
contains an n-term increasing subsequence or an n-term nondecreasing
subsequence; quantitatively, the smallest N with this property equals
(n-1)^2+1. In the setting of the present paper, we express this lemma by saying
that the set of predicates Phi={x_1<x_2,x_1\ge x_2}$ is Erdos-Szekeres with
Ramsey function ES_Phi(n)=(n-1)^2+1.
In general, we consider an arbitrary finite set Phi={Phi_1,...,Phi_m} of
semialgebraic predicates, meaning that each Phi_j=Phi_j(x_1,...,x_k) is a
Boolean combination of polynomial equations and inequalities in some number k
of real variables. We define Phi to be Erdos-Szekeres if for every n there
exists N such that each N-term sequence a of real numbers has an n-term
subsequence b such that at least one of the Phi_j holds everywhere on b, which
means that Phi_j(b_{i_1},...,b_{i_k}) holds for every choice of indices
i_1,i_2,...,i_k, 1<=i_1<i_2<... <i_k<= n. We write ES_Phi(n) for the smallest N
with the above property.
We prove two main results. First, the Ramsey functions in this setting are at
most doubly exponential (and sometimes they are indeed doubly exponential): for
every Phi that is Erd\H{o}s--Szekeres, there is a constant C such that
ES_Phi(n) < exp(exp(Cn)). Second, there is an algorithm that, given Phi,
decides whether it is Erdos-Szekeres; thus, one-dimensional
Erdos-Szekeres-style theorems can in principle be proved automatically.Comment: minor fixes of the previous version. to appear in Duke Math.