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Solutionstosystemsofbinomialequations
9
Sincetheexponentsheremaybenegative,itisnaturaltorequireeachxżto
benonzero,i.e.,xż∈C∗=C\{0}foreachź=1,...,n.Sothevariablestake
valuesin(C∗)nwhichhasanaturalgroupstructuregivenbythecomponent-
wisemultiplication
(x1,...,xn)•(g1,...,gn):=(x1·g1,...,xn·gn).
Thosenotationsareparticularlyconvenient,sincethefamiliaridentitiessuch
as
(2)
(x•y)A=xA•yAand(xA)B=xAB
stillhold.ALaurentpolynomialisalinearcombinationofLaurentmono-
mials,i.e.,anexpressionoftheformΣ
m
k=1ckxα
(k)
whereeachck∈Cand
α(k)∈Zn.ThesetofallsuchLaurentpolynomialsnaturallyformsaringun-
dertheusualpolynomialadditionandmultiplication.ALaurentbinomial
isaLaurentpolynomialhavingexactlytwotermswithnonzerocoefficients1,
i.e.,itisanexpressionoftheformc1xα+c2xβforsomec1,c2∈C∗and
α,β∈Zn.Eventhoughmuchoftheestablishedtheoryiswidelyapplicable
toLaurentbinomialsoverarbitraryalgebraicallyclosedfields,inthisarticle,
however,ourattentionisrestrictedtothosewithcomplexcoefficients.The
focushereissolvingsystemsofLaurentbinomialsequations,orsimplyLau-
rentbinomialsystems,over(C∗)n.Moreformally,givenexponentvectors
α(1),...,α(m),β(1),...,β(m)∈Znandthecoefficientscż,j∈C∗,thegoalis
todescribethesetofallx∈(C∗)nthatsatisfiesthesystemofequations
(
I
I
4
I
I
l
c1,1xα
cm,1xα
(1)
(m)
+c1,2xβ
+cm,2xβ
(1)
(m)
=0
.
.
.
=0.
Concerningthesolutionsetin(C∗)n,thissystemisclearlyequivalentto
(xα
(1)1β(1),...,xα
(m)1β(m))=(−c
1,2/c1,1,...,−cm,2/cm,1).
Withmorecompact“matrixexponent”notationsin(1),thissystemcansim-
plybewrittenas
(3)
xA=b
1AnalternativedefinitionforaLaurentbinomialthatisoftenusedis“aLaurentpoly-
nomialwithatmosttwotermswithnonzerocoefficients”.Thisdefinitionwouldinclude
Laurentmonomialsasaspecialcase.However,sincemonomialequationsaretrivialtosolve
fromapurelycomputationalpointofview,herewechoosetousethemorestrictdefinition.
