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8
TianranChen,Tien-YienLi
Wewilltakeacomputationalpointofviewinthisarticletooutlinethe
differentingredientsneededtosolvebinomialpolynomialsystemsexactlyand
numerically.Tofacilitatethediscussion,Section2introducesnotationsand
conventionstobeused.Section3discussesthestructureofsolutionsetsof
binomialsystemsandtheirimportantproperties.Section4investigatesdiffer-
entaspectsinsolvingbinomialsystemsfromacomputationalpointofview.
Inparticular,wewilldiscusstheproblemsincomputationwhenthecoeffi-
cientsofthebinomialsystemsarenotprovidedexactly.Amongvariouskinds
ofapplicationsforbinomialsystems,wepresent,inSection5,twoimportant
andinterestingapplications.Firstoneillustratestheheavyrelianceonsolv-
ingbinomialsystemswhensolvinggeneralpolynomialsystemsbytheefficient
polyhedralhomotopies.Secondexamplehighlightstheneedsanddifficulties
insolvingbinomialsystemsthataroseinsupersymmetricgaugetheoryinthe-
oreticalphysics.Finally,forsolvinglargerbinomialsystemsfromapplied
sciences,parallelcomputationbecomesinevitable.Section6showcasessome
recentdevelopmentsintheparallelimplementationofsolversforbinomial
systems.
2.Notationsandconcepts
Throughoutthisarticle,vectorsaredenotedbyboldfaceletterswhilema-
tricesaredenotedbycapitalletters,andMn×m(Z)denotesthesetofalln×m
matricesofintegerentries.Asquareintegermatrixissaidtobeunimodular
ifitsdeterminantis±1;subsequently,aunimodularmatrixinMn×n(Z)must
haveaninversethatisalsoinsideMn×n(Z)byCramer}srule.
Thoughthefocusofthisarticleisthebinomialsystems,itisconvenient,
andalmostnecessary,toextendthescopeofourdiscussiontomoregeneral
Laurentbinomialsystemsgivenasfollows.Forx=(x1,...,xn)andaninteger
columnvectorα=(O1,...,On)T∈Zn,the“vectorexponent”notationis
commonlyusedfortheLaurentmonomial:
/
N
N
N
O1
.
.
.
\
N
N
N
xα=(x1,...,xn)
\
On
J
=x
o1
1
···xon
n.
Similarly,foranintegermatrixA∈Mn×m(Z)withcolumnsα(1),...,α(m)∈
Zn,the“matrixexponent”notationisusedforasystemofLaurentmonomials:
(1)
xA=x(α(1)···α(m)):=(xα(1),...,xα(m)).
