Table Of ContentFoundations of
Computational
Mathematics
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Foundations of
Computational
Mathemati
Proceedings of the SMALEFEST 2000
Hong Kong, 13 - 17 2000
editors
Felipe Cucker
City University of Hong Kong
J. Maurice Rojas
Texas A&M University
V^b World Scientific
WB New Jersey London 'SSiinngqaappoorree *• Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Proceedings of Smalefest 2000
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
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ISBN 981-02-4845-8
Printed in Singapore by World Scientific Printers
V
FOREWORD
In August 1990 a conference celebrating the 60th birthday of Steve Smale was
held at the University of California at Berkeley. The goal of that conference, in
the words of its organizers, was "to gather in a single meeting mathematicians
working in the many fields to which Smale has made lasting contributions."
Thus, the contributed and invited lectures covered a broad scope of subjects
including Differential Topology, Dynamical Systems, and Mathematical Eco
nomics, among many others. A volume containing most of those lectures was
subsequently published by Springer-Verlag (From Topology to Computation,
Proceedings of the Smalefest, M. W. Hirsch, J. E. Marsden, M. Shub (Eds.),
Springer-Verlag, 1993).
Steve moved to City University of Hong Kong in 1995 and on July 15th
2000 he turned 70. It was a pleasure for his friends and colleagues to orga
nize a conference to celebrate this event. On July 13-17, 2000, the second
Smalefest was held in Hong Kong. Unlike the first one, however, the goal
was to focus on the subject Steve had been working on since the early 80's:
Theory of Computation. It was a simple matter to gather people who had
been influenced by Steve's work on the Theory of Computation, and a glance
at this volume shows that other subjects were quite well represented as well.
In the the first Smalefest volume, articles were grouped according to sub
jects and each group of articles was preceded by an article commenting on
Steve's work on that subject. In this volume we have included one such arti
cle — "The Work of Steve Smale on the Theory of Computation: 1990-1999"
— doing so for the period between the two conferences. We thank Singapore
University Press and World Scientific which granted us permission to reprint
this article. For the remaining articles, we thank the contributors for their
valuable work.
Special thanks go to the referees, who helped us select and polish the
papers in this volume; to the Liu Bie Ju Centre for Mathematical Sciences for
its generous sponsorship; and to Ms. Robin Campbell for her lightning-fast
LaTeX formatting.
Steve Smale has positively influenced not only our mathematics but —
through his friendship, sincerity, and generosity — our lives. It is with great
pleasure that we (the editors and the contributors) dedicate this volume to
Steve Smale as a belated gift on his 70th birthday. Happy 70 Steve!
Felipe Cucker
J. Maurice Rojas
December 2001
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VII
CONTENTS
Foreword v
Extending Triangulations and Semistable Reduction 1
DAN ABRAMOVICH AND J. MAURICE ROJAS
The Work of Steve Smale on the Theory of Computation: 1990-1999 15
LENORE BLUM AND FELIPE CUCKER
Data Compression and Adaptive Histograms 35
OLIVIER CATONI
Bifurcations of Limit Cycles in Z -Equivariant Planar Vector Fields 61
q
of Degree 5
HENRY S. Y. CHAN, K. W. CHUNG, AND JIBIN LI
Systems of Inequalities and the Stability of Decision Machines 85
JEAN-PIERRE DEDIEU
Reconciliation of Various Complexity and Condition Measures for 93
Linear Programming Problems and a Generalization of Tardos'
Theorem
JACKIE C. K. HO AND LEVENT TUN$EL
On the Expected Number of Real Roots of a System of Random 149
Polynomial Equations
ERIC KOSTLAN
Almost Periodicity and Distributional Chaos 189
GONGFU LIAO AND LIDONG WANG
Polynomials of Bounded Tree-Width 211
JANOS A. MAKOWSKY AND KLAUS MEER
Polynomial Systems and the Momentum Map 251
GREGORIO MALAJOVICH AND J. MAURICE ROJAS
VIII
Asymptotic Acceleration of the Solution of Multivariate Polynomial 267
Systems of Equations
BERNARD MOURRAIN, VICTOR Y. PAN, AND OLIVIER RUATTA
IBC-Problems Related to Steve Smale 295
ERICH NOVAK AND HENRYK WOZNIAKOWSKI
On Sampling Integer Points in Polyhedra 319
IGOR PAK
Nearly Optimal Polynomial Factorization and Rootfinding I: 325
Splitting a Univariate Polynomial into Factors over an Annulus
VICTOR Y. PAN
Complexity Issues in Dynamic Geometry 355
JURGEN RICHTER-GEBERT AND ULRICH H. KORTENKAMP
Grace-Like Polynomials 405
DAVID RUELLE
From Dynamics to Computation and Back? 423
MIKE SHUB
Simultaneous Computation of All the Zero-Clusters of a Univariate 433
Polynomial
JEAN-CLAUDE YAKOUBSOHN
Cross-Constrained Variational Problem and Nonlinear Schrodinger 457
Equation
JIAN ZHANG
1
EXTENDING TRIANGULATIONS AND SEMISTABLE
REDUCTION
D. ABRAMOVICH*
Department of Mathematics, Boston University
111 Cummington, Boston, MA 02215, USA
abrmovicQmath.bu.edu
http://math.bu.edu/INDIVIDUAL/abrmovic
J. M. ROJAS*
Department of Mathematics, Texas A&M University
College Station, TX 77843-3368, USA
roj asQmath.tamu.edu
http://www.math.tamu.edu/~roj as
1 INTRODUCTION
In the past three decades, a strong relationship has been established between
convex geometry, represented by convex polyhedra and polyhedral complexes,
and algebraic geometry, represented by toric varieties and toroidal embed-
dings. In this note we exploit this relationship in the following manner. We ad
dress a basic problem in algebraic geometry: a certain version of semistable
reduction.
Semistable reduction, for non-algebraic geometers, can be thought of as a
far-reaching extension of Hironaka's famous resolution of singularities 8.a
Technically, Hironaka's result is semistable reduction over a O-dimensional
base (see problem 1.3 below). Semistable reduction over a 1-dimensional
base was proved in 13, and was later applied in the classification of algebraic
threefolds 14 and the enumerative geometry of curves 4'5 to name but a few
examples. Semistable reduction for families of surfaces and threefolds (i.e.,
part of the case of a 2-dimensional base), in characteristic 0, was proved in n
but remains an open problem for a higher-dimensional base. This has moti
vated alternative constructions, e.g, weak semistable reduction (see theorem
'PARTIALLY SUPPORTED BY NSF GRANT DMS-9503276 AND AN ALFRED P.
SLOAN RESEARCH FELLOWSHIP.
tPARTIALLY SUPPORTED BY AN NSF MATHEMATICAL SCIENCES POSTDOC
TORAL FELLOWSHIP AND HONG KONG UGC GRANT #9040402-730.
"Roughly, his result is that any algebraic variety over an algebraically closed field of char
acteristic 0 is birationally equivalent to one without singularities.
2
1.6 below and the paragraph after the theorem), which could be proved in full
generality in characteristic 0 2, and has also yielded important applications
10,17
Here, we will translate the local case of semistable reduction, over a base
variety of dimension > 1, into a basic problem about polyhedral complexes:
extending triangulations. Once we solve the second problem, the first follows.
We have taken the opportunity with this note to try to extend some bridges
between the terminologies of these two theories.
1.1 Semistable Reduction
We work over the field of complex numbers C. Let / : X —> B be a proper
morphism of algebraic varieties, whose generic fiber is reduced and absolutely
irreducible. Thus there exists a Zariski dense open set U C B such that the
fiber /_1(6) over any point in b € U is a compact complex algebraic variety.
Loosely speaking, semistable reduction for a morphism like / is a meta-
problem of "desingularization of morphisms," where the goal is to "change
/ slightly" so that it becomes "as nice as possible". Of course, we need to
specify more precisely what we mean by the clauses in quotation marks.
1.1.1 What do we mean by a morphism being "as nice as possible?"
First of all, X and B should be as nice as possible, namely nonsingular.
Moreover, we want / to have a nice, explicit local description, so that the
fibers of / have the simplest possible singularities.
Such a morphism will be called semistable. Here is the definition:
Definition 1.1 Let f : X —>• B be a flat projective morphism, with con
nected fibers, of nonsingular varieties. We say that f is semistable if for
each point x G X with f(x) = b there is a choice of formal coordinates
Bf, = Spec C[[ti,... ,t }] and X = Spec <C[[a;i,... ,£„]], such that f is
m x
given by:
h
ti = Y\_ xj:
j=Ji_i+i
where 0 = IQ < h • • • < l < n, n = dimX, and m = dimB.
m
We must state right up front that in this note we will not end up with
a semistable morphism, but we will get very close. In particular, our results
form an additional step in work on semistable reduction x, continuing the
work of 2 for the case of a higher-dimensional base.
