Preface
This boo k gre w ou t o f th e note s fo r te n lecture s give n b y th e autho r a t th e
CBMS Conferenc e a t Texa s A& M University , Colleg e Station , durin g th e wee k o f
May 20-24, 2002. Paul o Lima Filho, J. Maurice Rojas an d Hal Schenck did a fantas-
tic job o f organizing thi s conferenc e an d takin g car e o f mor e tha n 8 0 participants ,
many o f the m graduat e student s workin g i n a wid e rang e o f mathematica l fields .
We were fortunate t o be abl e to listen t o th e excellen t invite d lecture s delivere d b y
the following twelve leading experts: Saugat a Basu, Eduard o Cattani , Karin Gater -
mann, Crai g Huneke , Tien-Yie n Li , Gregori o Malajovich , Pabl o Parrilo* , Mauric e
Rojas, Fran k Sottile , Mik e Stillman* , Thorste n Theobald , an d Ja n Verschelde* .
Systems o f polynomia l equation s ar e fo r everyone : fro m graduat e student s
in compute r science , engineering , o r economic s t o expert s i n algebrai c geometry .
This boo k aim s t o provid e a bridg e betwee n mathematica l level s an d t o expos e a s
many facet s o f the subjec t a s possible . I t cover s a wide spectru m o f mathematica l
techniques an d algorithms , bot h symboli c an d numerical . Ther e ar e tw o chapter s
on applications . Th e on e abou t statistic s i s motivate d b y th e author' s curren t
research interests , an d th e on e abou t economic s (Nas h equilibria ) recognize s Dav e
Bayer's rol e i n th e makin g o f th e movi e A Beautiful Mind. (Man y thanks , Dave ,
for introducin g m e to th e star s a t thei r kick-of f part y i n NYC o n Marc h 16, 2001.)
At th e en d o f eac h chapte r ther e ar e abou t te n exercises . Thes e exercise s
vary greatl y i n thei r difficulty . Som e ar e straightforwar d application s o f materia l
presented in the text while other "exercises " ar e quite hard and ought to be renamed
"suggested researc h directions" . Th e reade r ma y decid e whic h i s which .
We ha d a n inspirin g softwar e sessio n a t th e CBM S conference , an d th e jo y
of computin g i s reflecte d i n thi s boo k a s well . Sprinkle d throughou t th e text ,
the reade r find s shor t compute r session s involvin g polynomia l equations . Thes e
involve the commercial packages Maple® andMATLAB® as well as the freely availabl e
packages Singular
1,
Macaula y 2
2,
PHCpack
3,
an d SOSTools
4.
Developer s o f th e
last thre e program s spok e a t th e CBM S conference . Thei r name s ar e marke d wit h
a sta r above .
There ar e man y fin e compute r program s fo r solvin g polynomia l system s othe r
than th e one s liste d above . Sadly , I di d no t hav e tim e t o discus s the m all . On e
1
Singular i s a free software distribute d unde r the GN U license . © Departmen t o f Mathematic s
and Centr e fo r Compute r Algebra , Universit y o f Kaiserslautern , German y
2
Macaulay2: ©Danie l R . Grayso n an d Michae l E . Stillma n (1993-2001 ) an d i s distribute d
free unde r th e GN U licens e
3PHCpack: © 1998, Katholieke Universitei t Leuven , Departmen t o f Compute r Science , Hev -
erlee, Belgiu m
4SOSTools i s a MATLAB ® toolbo x an d freel y availabl e unde r th e GN U licens e at :
http://www.cds.caltech.edu/sostools o r http://www.aut.ee.ethz.ch/~parrilo/sostool s
vii
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