Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
Solutionstosystemsofbinomialequations
11
Weshallnotincludetheproofofthispropositionhere,asitissubsumed
inthemoreconcretedescriptionofthesolutionsetinProposition2.(Also,
thispropositioncanbeconsideredasacorollaryof[12,Theorem2.1]when
thetheoremisappliedtoLaurentbinomialsystemsdefinedoverC.)
ForPandQintheSmithNormalFormofAin(4),letPr∈Mr×n(Z)
andPo∈M(n1r)×n(Z)bethetoprrowsandtheremainingn−rrowsof
Prespectively.Similarly,letQr∈Mm×r(Z)andQo∈Mm×(m1r)(Z)bethe
leftrcolumnsandtheremainingm−rcolumnsofQrespectively.Withthese
notations,theSmithNormalFormofAcanbewrittenas
(5)
(Pr
Po)A(QrQo)=(D0
0
0)
withD=diag(d1,...,dr)∈Mr×r(Z)and0)srepresentingzeroblockmatrices
ofappropriatesizes.
ThesquarematrixQin(4)inducesamapyl→yQ.Thismapisan
automorphismof(C∗)m,sinceQisunimodular(andhencehasaninversein
Mm×m(Z)).Thus,consideringthesolutionsetin(C∗)n,theoriginalLaurent
binomialsystemxA=bisequivalentto
(xA)Q=xAQ=bQ.
Similarly,sincePin(4)isaunimodularn×nmatrix,themapzl→zPis
alsoanautomorphismof(C∗)n.Sothesolutionsetsremainequivalentafter
thechangeofvariablesx=zP,andtheLaurentbinomialsystembecomes
(zP)AQ=zPAQ=z(D0
00)=(z(
D
0),z(
0
0))=bQ=(bQr,bQ0).
SinceD=diag(d1,...,dr)∈Mr×r(Z),theoriginalsystemxA=bcannow
bedecomposedintoacombinedsystem
(6)
(7)
(8)
(z1,...,zr)
/
N
\
d1
...
dr
\
N
J
=b
Qr
1=b
Q0
zr+1,...,zn:
free
where(7)appearswhenr<mwith1=(1,...,1)∈(C∗)m1r,and(8)
appearswhenr<n.Theword“free”in(8)meansthesystemimposesno
constraintsonthen−rvariableszr+1,...,zn.
