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AnnalesMathematicaeSilesianae28(2014),7-34
PraceNaukoweUniwersytetuŚląskiegonr3240,Katowice
SOLUTIONSTOSYSTEMSOFBINOMIALEQUATIONS
TianranChen,Tien-YienLi
AnnualLecturededicatedtothememoryofProfessorAndrzejLasota
1.Introduction
Theproblemofsolvingsystemsofpolynomialequationshasbeen,and
willcontinuetobe,oneofthemostimportantsubjectsinbothpureand
appliedmathematics.Theneedtosolvesystemsofpolynomialequationsoc-
cursfrequentlyinvariousfieldsofscienceandengineering,suchasformulae
construction,geometricintersection,inversekinematics,robotics,computer
vision,andthecomputationofequilibriumstatesofchemicalreactionequa-
tions.
Inthisarticle,wewillfocusonsolvingbinomialpolynomialsystems:
Systemsofpolynomialequationsinwhicheachequationcontainsexactlytwo
terms.Binomialpolynomialsystems(orsimplybinomialsystems)repre-
sentsanimportantclassofpolynomialsystems.Whilethesolutionstructure
ofbinomialsystemsisinterestinginitsownright,itactuallyplaysacrucially
importantroleinsolvinggeneralpolynomialsystemsnumerically.(Wewill
elaboratethisindetailsinSection5.1.)Moreover,binomialsystemshave
directconnectionstolatticeideals[34]andtoricvarieties[9,15],whichadmit
vastapplications.
Received:22.03.2014.Revised:6.08.2014.
(2010)MathematicsSubjectClassification:14Q99,14M25.
Keywordsandphrases:binomialsystem,BKKbound,masterspace,polyhedral
homotopy.
ResearchsupportedinpartbyNSFunderGrantDMS11-15587andbyNSFCunder
Grant11171052.
