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TianranChen,Tien-YienLi
wheretheintegermatrixA∈Mn×m(Z),havingcolumnsα(1)−β(1),...,α(m)−
β
(m),representstheexponentsappearedintheLaurentmonomialsandthe
vectorb=(−c1,2/c1,1,...,−cm,2/cm,1)∈(C∗)mcollectsallthecoefficients.
Inthisarticle,weoftentake(3)tobethestandardformofaLaurentbino-
mialsystem,withwhichonemustbearinmindthatforA∈Mn×m(Z),n
representsthenumberofvariablesandmrepresentsthenumberofbinomial
equationsinthesystem.
3.SolutionsetsofLaurentbinomialsystems
Inthissection,weoutlinethestructuraltheoryofthesolutionsetofa
Laurentbinomialsystemaswellasthemeansbywhichonecouldstudythe
importantpropertiesofthesolutionsetwithregardtoitsdimension,number
ofcomponents,smoothness,degree,andglobalparametrizations.Thedetails
ofamoregeneraltheorycanbefoundin[9],[12],[15],and[34].
Animportanttoolinunderstandingthestructureofthesolutionsetof
aLaurentbinomialsystemxA=b(initsstandardform(3))istheSmith
NormalFormoftheexponentmatrixA:Itisknownthatforintegermatrix
A,thereareunimodularsquarematricesP∈Mn×n(Z)andQ∈Mm×m(Z)
suchthat
r
m−r
PAQ=
(
|
|
|
|
|
|
|
|
|
\
d1
...
dr
0
...
0
)
\
|
|
|
|
|
|
|
|
|
r
n−r
(4)
withnonzerointegersd1|d2|···|drforr=rankA,uniqueuptothesigns.
Here,a|bmeansadividesbasusual.ThisdecompositionofthematrixA
providesimportantgeometricinformationaboutthesolutionsetofxA=b
in(C∗)nsummarizedinthefollowingproposition:
Proposition1.IfthesolutionsetofxA=bin(C∗)nisnotempty,then
itconsistsofafinitenumberofconnectedcomponents.Furthermore,
1.thenumberofsolutioncomponentsisexactly|
|Π
r
j=1dj
|
|;
2.eachsolutioncomponenthascodimensionequalsrankA=r;and
3.eachsolutioncomponentissmooth.
