Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

arXiv:0904.2667v2 [math.CV] 5 Nov 2009

o tonioni variable: a division lemma and the

amshaft ee t R. Ghiloni,A. Perotti

∗

Department of Mathemati s University of Trento ViaSommarive,14 I38123 PovoTrento ITALY

perottis ien e.unitn.it November5, 2009

Abstra t

Westudyindetailthezerosetofasli eregularfun tionofa quater-nioni oro tonioni variable. Bymeansofadivisionlemmafor onvergent powerseries,wendtheexa trelationexisting betweenthezerosoftwo o tonioni regular fun tions and those of their produ t. In the ase of o tonioni polynomials,wegetastrongformofthefundamentaltheorem ofalgebra. Weprovethatthesumofthemultipli itiesofzerosequalsthe degreeofthepolynomialandobtainafa torizationinlinearpolynomials. Keywords: Quaternions, O tonions, Fun tions of ahyper omplex variable, Fundamentaltheoremofalgebra

Math.Subj.Class: 30C15,30G35,32A30

1 Introdu tion

Let

H

denotetheskeweldofquaternionsover

R

. Re ently,GentiliandStruppa [13,15℄introdu edthenotionofsli eregularity (orCullen regularity)for fun -tionsofaquaternioni variable. Thisnotiondonot oin idewith the lassi al oneof Fueter regularity(we referthereader to [30℄ and[19℄for thetheory of Fueter regular fun tions). In fa t, the set of sli e regular fun tions on a ball

B

R

enteredin the originof

H

oin ides withthat of allpowerseries

P

i

w

i

a

i

that onverges in

B

R

. In parti ular, all the standardpolynomials

P

n

i=0

w

i

a

i

withrightquaternioni oe ients,whi h failtobeFueter regular,dene sli e

∗

WorkpartiallysupportedbyMIUR(PRINProje tProprietàgeometri hedellevarietà realie omplesse")andGNSAGAofINdAM

regular fun tions on the whole spa e

H

. The lass of sli e regular fun tions ontainsalsothe radiallyholomorphi fun tions introdu ed byFueter ( f. [19, Ÿ11℄). Su h fun tions x every omplex plane generated by the reals and by aunit imaginaryquaternion, while the sli e regular fun tions do not need to fullthis property. A tually,radiallyholomorphi fun tions orrespondto the veryspe ial lassof those sli eregularfun tions, whi h anbeexpanded into powerserieswith real oe ients. Thenotionofsli eregularitywasextended tofun tionsofano tonioni variablein[14,12℄andtofun tionswithvaluesin aCliordalgebrain[17, 4℄.

Inthispaper,westudyin detailthezerosetofasli eregularfun tion. For simpli ity, in thesequel,weuse theterminologyregular fun tion in pla e of sli eregularfun tion. Weobtainby dierent te hniques some properties al-readyknown( f.[24,14,15,11,16,18℄)andweextendotherstotheo tonioni ase. We nd the exa trelation existing between the zerosof twoo tonioni regular fun tions and those of their produ t. In the ase of o tonioni poly-nomials,weobtainastrongformof thefundamental theoremof algebra. The main tool we use is a division lemma, whi h generalizes to o tonioni power seriesaresultprovedbyBe k[1℄ forquaternioni polynomialsand by Serdio [27℄foro tonioni polynomials. Inthesimpler aseofregularpolynomialsand onvergentpowerserieswithreal(i.e. entral) oe ients,thepropertiesofthe zeroset havebeenstudied alsobyotherauthors( f. [5℄).

We des ribe in more details the stru ture of the paper. In Se tion 2, we re allsomebasi denitions and provethedivision lemma (Lemma 1). Given a regular fun tion

f

and an o tonion

α

, we an asso iate to

f

a remainder, whi h is a linear or onstant regular polynomial. This remainder des ribes ompletelythe interse tion ofthe zeroset

V (f )

of

f

withthe onjuga y lass of

α

. We an then obtaineasily a stru ture theorem for

V (f )

(Theorem 6). This result was proved for quaternioni polynomials by Pogorui and Shapiro [24℄, for quaternioni regular fun tions byGentili and Stoppato in [11℄ and it wasextendedto o tonioni regularfun tions in [18℄. Thedivision lemma and the on eptofnormal series asso iatedtoaregularfun tion (seeSe tion2for pre isedenitions)suggestthedenitionofthemultipli ityofazeroofaregular fun tion. Ourdenitionisequivalentonquaternioni polynomialswiththeone givenin[2℄andin[16℄.

InSe tion3,wedes ribethezerosoftheprodu t

f ∗g

oftworegularfun tions intermsofthezerosof

f

and

g

. Weprove(Lemma7)thatthenormalseriesof aregularfun tionismultipli ative,aresultthatisnontrivialintheo tonioni ase. An immediate onsequen eis that the onjuga y lassesof the zerosof

f ∗ g

areexa tlythose ontainingzerosof

f

or

g

. Thepre iserelationbetween

su hzerosisstatedinTheorem9. We allthisrelationthe amshaftee t. In theasso iative ase,itredu estoknownresults( f.[11℄forquaternioni regular fun tionsand[22,29℄forpolynomials).

Se tion4 ontainssomeappli ationsofthepre edingresultstoregular poly-nomials. The fundamental theorem of algebra was proved by Niven [23℄ for standardquaternioni polynomials. Itwasextended to otherpolynomialsover

theorem valid for a lass of real algebras in ludingthe omplex numbers, the quaternionsandtheo tonions. Seealso[31℄,[25℄and[18℄ forotherproofs.

In this ontext, our aim is a strong form of the fundamental theorem, in whi h aformulafor thesumof themultipli ities of zerosisa hieved. Gordon andMotzkin[20℄ proved,forpolynomialsona(asso iative)division ring,that thenumberof onjuga y lasses ontainingzerosof

f

annot be greaterthan thedegree

n

of

f

. This estimatewasimprovedonthequaternions by Pogorui and Shapiro [24℄: if

f

has

m

spheri al zeros and

l

nonspheri al zeros, then

2m + l ≤ n

. Gentili and Struppa [16℄ showed that, using the right denition

of multipli ity, the number of zeros of

f

equals the degreeof the polynomial. Wegeneralize this strong form to theo tonions (Theorem 11), givingaproof inspiredbythesimpleargumentusedin[24℄. Fromthisresultandthedivision lemma,wegetafa torizationlemmaforregularpolynomials(Lemma12)and somesu ient onditionsforthenitenessof

V (f )

.

Thedenitions andresultsarestatedovero tonions only, buttheyremain validoverquaternionsaswell. Sin e

H

anbeidentiedwitharealsubalgebra oftheo tonions,the orrespondingstatementsforquaternions anbeobtained dire tlyfromtheiro tonioni versionbyspe ializing oe ientsandvariables.

2 Division of regular series and multipli ity of

zeros

Let

O

bethenonasso iativedivision algebraofo tonions (also alled Cayley numbers). Wereferto[26,22,6℄forthemainpropertiesofthealgebras

H

and

O

. Were allthat

O

anbeobtainedfrom

H

bytheCayleyDi ksonpro ess. Any

x ∈ O

an be written as

x = x

1

+ x

2

k

, where

x

1

, x

2

∈ H

and

k

is a xed

imaginary unit of

O

. The addition on

O

is dened omponentwise and the produ tisdenedby

xy = (x

1

+ x

2

k)(y

1

+ y

2

k) = x

1

y

1

− ¯

y

2

x

2

+ (x

2

y

¯

1

+ y

2

x

1

)k.

Let

{1, i, j, ij}

denote arealbasis of

H

. Then abasis ofthe real algebra

O

is

formedby

{1, i, j, ij, k, ik, jk, (ij)k}

.

Let

B

R

betheopenballof

O

enteredin theoriginwith(possiblyinnite) radius

R

. Let

f : B

R

−→ O

and

g : B

R

−→ O

be regular fun tions with seriesexpansions

f (w) =

P

i

w

i

a

i

and

g(w) =

P

j

w

j

b

j

. Theusual produ tof polynomials,where

w

is onsideredtobea ommutingvariable( f.forexample [22℄ and [9, 10℄), an be extended to power series, and hen e to sli e regular fun tions( f. [11,16℄). Re allthat

f ∗ g

is theregular fun tiondened on

B

R

bysetting

(f ∗ g)(w) :=

P

k

w

k

P

i+j=k

a

i

b

j

.

We denote by

f

the regular fun tion dened on

B

R

by

f (w) :=

P

i

w

i

a

i

. If

(f ∗ g)(α) = f (α)g(α)

forevery

g

andevery

α ∈ O

andwewillwrite

f g

inpla e

of

f ∗ g

. Moreover,if

f

isreal then

(f ∗ g) ∗ h = (f g) ∗ h = f (g ∗ h)

for every

g, h

,and

f ∗ g

isequalto

f g

.

Wedenoteby

N (f )

thenormalseries (or symmetrizedseries)of

f

dened by

N (f ) := f ∗ f = f ∗ f.

Weremark that

N (f )

isregularandrealon

B

R

.

Let

α ∈ O

. Wedenotethetra e

tr(α) = α + α

of

α

alsoby

t

α

,thesquared

normof

α

by

n

α

anddenetherealpolynomial

∆

α

, alled hara teristi poly-nomialof

α

, by

∆

α

(w) := N (w − α) = (w − α) ∗ (w − α) = w

2

− w · t

α

+ n

α

.

It is wellknown ( f. [27℄) that

α

is onjugate to an o tonion

β

if and only if

∆

α

= ∆

β

. Indi ate by

S

α

the onjuga y lass of

α

:

S

α

:= {β ∈ O | β =

Re(α) + |Im(α)| I, I ∈ S}

, where

S = {I ∈ O | I

2

_{= −1}}

is the sphere of imaginaryunits. Notethat

S

α

redu estothepoint

{α}

when

α

isrealanditis asixdimensionalspherewhen

α

isnonreal.

Wedenote by

V (f )

thezero set of

f

. If

α ∈ V (f )

is real, we all

α

areal zero of

f

. If

α

is nonreal and

S

α

⊂ V (f )

, we all

α

a spheri al zero of

f

. Otherwise,we all

α ∈ V (f )

anisolatedzeroof

f

. Thisterminologyisjustied byTheorem 6below.

Letuspresentourdivisionlemma.

Lemma1. Let

f : B

R

−→ O

regularandlet

α ∈ B

R

. Thefollowingstatements hold.

(1)

There exist, and are unique, a regular fun tion

g : B

R

−→ O

and an o tonion

r ∈ O

su hthat

f (w) = (w − α) ∗ g(w) + r

.

(2)

Let

α ∈ O

be nonreal. There exist, and are unique, a regular fun tion

h : B

R

−→ O

ando tonions

a, b ∈ O

su hthat

f (w) = (∆

α

h)(w)+w a+b

.

Proof. (1)Letusprovetheexisten eof

g

andof

r

. First,observethat,ifsu h

g

and

r

exist,then

r

mustbeequalto

f (α)

. Infa t,it iseasyto see that the valueofthefun tion

(w − α) ∗ g(w)

in

α

iszero. Assume that

f =

P

i

w

i

a

i

has positive onvergen eradius

R

. Re allthat

R =



lim sup

n→+∞

n

p|a

n

|



−1

.

Let

α ∈ B

R

. If

α = 0

,thenwe anwrite

f = w



P

+∞

i=0

w

i

a

i+1



+ a

0

.

Suppose

α 6= 0

. A formal series omputation imposes to dene the oe ients of the

powerseries

g =

P

n

w

n

b

n

asfollows:

b

n

:= α

−1−n



f (α) −

n

X

j=0

α

j

_a

j





= α

−1−n

+∞

X

j=n+1

α

j

_a

j

forea h

n ∈ N

. Letusshowthattheseries

g

hasradiusof onvergen eatleast

R

. Sin e

|α| < R

,

lim sup

n→+∞

p|a

n

n

| < |α|

−1

. Thisimpliesthat,foreveryxed

ρ

with

|α| < ρ < R

,there existsaninteger

n

ρ

su hthattheinequality

|a

j

| ≤ ρ

−j

issatisedforevery

j > n

ρ

. Let

x ∈ B

R

andlet

ρ

be hoseninsu hawaythat

max{|α|, |x|} < ρ < R

. Then,forevery

n ≥ n

ρ

,itholds:

|x

n

_b

n

| ≤ |x|

n

|b

n

| ≤ |x|

n

|α|

−1−n

+∞

X

j=n+1

|α|

j

_|a

j

| ≤ |x|

n

|α|

−1−n

+∞

X

j=n+1

|α|

j

_ρ

−j

= |x|

n

|α|

−1−n

 |α|

ρ



n+1

1

1 − |α|/ρ

=

 |x|

ρ



n

1

ρ − |α|

.

Therefore, the power series

P

n

w

n

b

n

onverges at every

x ∈ B

R

. It remains to prove the uniqueness of

g

and of

r

. We have just seen that

r

must be equal to

f (α)

so the o tonion

r

is uniquely dened by

f

and

α

. If

f (w) =

(w − α) ∗ g(w) + r = (w − α) ∗ g

′

_{(w) + r}

,then

(w − α) ∗ (g(w) − g

′

_{(w)) = 0}

and thiseasilyimpliesthat

g = g

′

.

(2)Let

α

benonreal. Usingtherstparttwi e,weget

s ∈ O

andaregular fun tion

h

on

B

R

su hthat

f (w) = (w − α) ∗ g(w) + r = (w − α) ∗ ((w − ¯

α) ∗ h(w) + s) + r

= (∆

α

h)(w) + ws − αs + r.

To proveuniqueness, assume that

f (w) = ∆

α

h + wa + b = ∆

α

h

′

_{+ wa}

′

_{+ b}

′

.

Then

f (α) = αa + b = αa

′

_{+ b}

′

,

f (¯

α) = ¯

αa + b = ¯

αa

′

_{+ b}

′

, from whi h weget

(α − ¯

α)(a − a

′

_{) = 0}

andthen

a = a

′

,

b = b

′

. Finally,

∆

α

h = ∆

α

h

′

gives

h = h

′

on

B

R

\ S

α

. Byadensityargument,

h

and

h

′

must oin ideeverywhere. Remark 1. As abyprodu t ofthe proofof thepre eding theorem, weobtain that,if

f

isaregularo tonioni polynomialofdegree

n > 0

,then

g

isaregular o tonioni polynomialofdegree

n − 1

.

Denition1. Lettheregularfun tions

g, h

andtheo tonions

r, a, b

beasabove. If

α

is real, we all

r

the remainderof

f

withrespe t to

α

andthe fun tion

g

the quotientof

f

w.r.t.

α

. If

α

isnonreal,we all remainderof

f

withrespe t to

α

theo tonioni polynomialdenedby

w a + b

and quotientof

f

w.r.t.

α

the fun tion

h

. Notethat theydependonly onthe onjuga y lass

S

α

of

α

. In any ase,wewill denote theremainder by

r

α

(f )

.

If

f

isreal,thenthe oe ientsofthequotientfun tionandoftheremainder arereal. Thisfa tisa onsequen eoftheuniquenessofquotientandremainder:

(w − α) g + r = f = ¯

f = (w − α) ¯

g + ¯

r

(

α

real)

or

∆

α

h + w a + b = f = ¯

f = ∆

α

h + w ¯

¯

a + ¯b

(

α

nonreal)

imply

g = ¯

g

,

r = ¯

r

,

h = ¯

h

,

a = ¯

a

and

b = ¯b

. Asarstappli ation ofLemma1,weobtain:

Corollary2. Let

r

α

(f )

betheremainder of

f

withrespe tto

α

. Thefollowing statementshold.

(1)

Let

α

be real. Then

α

is a zero of

f

if and only if

r

α

(f ) = 0

; that is,

w − α | f

.

(2)

Let

α

benonreal andlet

r

α

(f )(w) = w a + b

. Then wehave:

(i) α

isaspheri al zeroof

f

if andonlyif

a = 0 = b

;thatis,

∆

α

| f

.

(ii) α

isanisolatedzeroof

f

if andonlyif

a 6= 0

and

α = −ba

−1

. In parti ular, either

S

α

⊂ V (f )

or

S

α

∩ V (f )

is empty or

S

α

∩ V (f )

onsists ofasinglepoint. Thelattersituation o urs ifandonly if

a 6= 0

and

−ba

−1

_{∈ S}

α

.

Proof. By using Lemma 1, we an follow the same lines of proof as in the polynomial ase( f. [27,Ÿ3℄).

If

α ∈ R

,thestatementistrivial,sin e

f (α) = r

α

(f )

.

Now assumethat

α

is nonreal. If

r

α

(f ) = 0

,i.e.

a = b = 0

, then

f (β) =

∆

α

(β) · h(β) = 0

for every

β ∈ S

α

. If

a 6= 0

and

α = −ba

−1

, then

f (α) =

∆

α

(α) · h(α) = 0

. If

S

α

ontainstwozeros

β

and

γ

of

f

,thentheequality

∆

α

(β) · h(β) + β a + b = f (β) = 0 = f (γ) = ∆

α

(γ) · h(γ) + γ a + b

gives

β a = γ a

, sin e

∆

α

vanishes on

β, γ ∈ S

α

. If

β 6= γ

, then it must be

a = b = 0

.

If

f

isreal,then

r

α

(f ) = w a + b

with

a, b

real. Therefore

−ba

−1

_{∈ R}

and

f

hasnoisolated(nonreal)zeros. Thismeansthat thezerosof

f

are allrealor spheri aland

V (f )

istheunionofthespheres

S

α

with

α ∈ V (f )

( f.[5℄).

If

f

diersfrom areal

g

bya onstantnonrealo tonion,thenwehavethe opposite situation:

f

an have only isolatedzeros. This property was proved forregularpolynomialsover

H

in[22, Ÿ16.19℄.

Proposition 3. Let

f (w) = g(w) + a

0

be aregular fun tion on

B

R

. Assume that

g

isreal and

a

0

∈ O

isnonreal. Then, if

V (f )

isnotempty, itselements areisolatedzeros of

f

ontainedin the omplex planegeneratedby

1

and

a

0

. Proof. Let

α ∈ V (f )

. Sin e

a

0

= −g(α)

is nonreal,also

α

must benonreal. From thedivision lemma (Lemma 1) applied to

g

, weget:

g(w) = ∆

α

h(w) +

wa + b

, with real

a, b

. Then

r

α

(f ) = wa + b + a

0

and

α

annotbe spheri al.

Otherwise, it would be

a

0

= −b

. Therefore,

α = −(b + a

0

)a

−1

is anisolated zeroof

f

belongingtothe omplexplanegeneratedby

1

and

a

0

.

We still get the nonexisten e of spheri al zeros under a weaker ondition on

f

.

Proposition 4. Let

f (w) = g(w) + wa

1

+ a

0

be a regular fun tion on

B

R

. Assumethat

g

isreal andeither

a

0

or

a

1

isnonreal. Then

f

has no spheri al zeros.

Proof. We pro eed as before. Assume that

V (f ) 6= ∅

. Let

α ∈ V (f )

. Then

r

α

(f ) = w(a+a

1

)+b+a

0

,with

a, b

real. Then

α

annotbespheri al.Otherwise,

itwouldbe

a

0

= −b

and

a

1

= −a

.

If

α

is real and

f (w) = (w − α) g(w) + r

forsome regular fun tion

g

and

r ∈ O

,then

r

α

(f ) = r

,

r

α

( ¯

f ) = ¯

r

and

N (f ) = (w − α)

2

N (g) + (w − α)(g ∗ ¯

r + r ∗ ¯

g) + n

r

.

(1)

Itfollowsthat

r

α

(N (f )) = n

r

. If

α

isnonrealand

f

hasremainder

r

α

(f )(w) =

w a + b

,thenwehavethat

r

α

( ¯

f )(w) = w ¯

a + ¯b

andhen e

r

α

(N (f )) = r

α

(f ∗ ¯

f )(w) = w(a¯b + b¯

a + t

α

n

a

) + n

b

− n

α

n

a

.

Thelatterequalityfollowsimmediatelyfromthefollowingfa t

(wa + b) ∗ (w¯

a + ¯b) =

w

2

n

a

+ w(a¯b + b¯

a) + n

b

=

=

(∆

α

+ wt

α

− n

α

)n

a

+ w(a¯b + b¯

a) + n

b

.

Corollary5. Given

α ∈ O

,the followingstatements areequivalent:

(1) S

α

∩ V (f )

isnonempty.

(2) S

α

∩ V (N (f ))

isnonempty.

(3) ∆

α

| N (f )

.

Inparti ular, wehave that

V (N (f )) =

[

α∈V (f )

S

α

.

Proof. If

α

isreal and

r

α

(f ) = r

,then

r

α

(N (f )) = n

r

= 0

ifandonlyif

r = 0

. If this is the ase, Eq. (1) implies that

∆

α

= (w − α)

2

divides

N (f )

. If

α

is nonreal,then

∆

α

| N (f )

ifandonlyif

r

α

(N (f )) = 0

;thatis,

a¯b + b¯

a + t

α

n

a

= 0,

n

b

− n

α

n

a

= 0.

(2)

If

a = b = 0

, then

S

α

⊆ V (f )

and equations (2) are trivial. If

a 6= 0

and

β = −ba

−1

_{∈ S}

α

∩ V (f )

,then

t

α

n

a

= t

β

n

a

= βa¯

a + a¯

a ¯

β = −b¯

a − a¯b

. Moreover,

n

α

n

a

= n

β

n

a

= n

b

andequations(2)aresatised.

Conversely,supposethatequations(2)aresatised. If

a = 0

,then

b = 0

and hen e

S

α

⊂ V (f )

. Let

a 6= 0

. Dene theo tonion

β := −ba

−1

. Pro eeding as above,weobtainthat

t

α

n

a

= t

β

n

a

and

n

α

n

a

= n

β

n

a

or,equivalently,

t

α

= t

β

and

n

α

= n

β

. Itfollowsthat

β ∈ S

α

∩ V (f )

. Sin e

N (f )

isreal,itszerosareall realorspheri al,so

V (N (f ))

isequalto

S

α∈V (f )

S

α

.

Fromthepre eding orollaries,weimmediatelygetanewproofofthe follow-ingresult. Itwasprovedforquaternioni polynomialsbyPogoruiandShapiro [24℄, for quaternioni regular fun tions byGentili and Stoppato in [11℄ and it wasextendedtoo tonioni regularfun tionsin [18℄.

Theorem6. Let

f : B

R

−→ O

bearegularfun tion,whi hdoesnotvanishon the whole

B

R

. For ea h

I ∈ S

, denote by

C

I

the omplexplane of

O

generated

by

1

and

I

; that is,

C

I

:= {x + yI ∈ O | x, y ∈ R}

. The following statements

hold.

(1)

Forall

I ∈ S

,

C

I

∩

S

α∈V (f )

S

α

is losedanddis retein

C

I

∩ B

R

.

(2)

For ea h

β ∈

S

α∈V (f )

S

α

, either

S

β

⊂ V (f )

or

S

β

∩ V (f )

onsists of a singlepoint.

Proof. Sin e

N (f )

is regular on

B

R

, and then holomorphi on

C

I

∩ B

R

, its zeroset

C

I

∩

S

α∈V (f )

S

α

= C

I

∩ V (N (f ))

is losed anddis rete in

C

I

∩ B

R

. These ond statementfollowsfromCorollary2.

Denition2. Given anonnegativeinteger

s

and ano tonion

α

in

V (f )

,we say that

α

is a zeroof

f

of multipli ity

s

if

∆

s

α

| N (f )

and

∆

s+1

α

∤ N (f )

. We

willdenote theinteger

s

, alled multipli ityof

α

,by

m

f

(α)

.

Inthe aseof

α

arealelement,this onditionis equivalentto

(w − α)

s

_{| f}

and

(w − α)

s+1

_{∤ f}

. If

α

isaspheri alzero,then

∆

α

divides

f

and

f

¯

. Therefore

m

f

(α)

is at least 2. If

m

f

(α) = 1

,

α

is alled asimple zero of

f

. The reader observesthatthemultipli ityof

α

dependsonlyonthe onjuga y lass

S

α

of

α

. Remark 2. Thepre eding denition isequivalenton quaternioni polynomials withtheonegivenin[2℄andin[16℄.

3 Zeros of produ ts: the  amshaft ee t

Let

f, g : B

R

−→ O

beregularfun tions. Theaimofthisse tionistodes ribe

the zeros of

f ∗ g

in terms of the zeros of

f

and of

g

. The following result is nontrivial andveryimportant in the o tonioni setting. It wasprovedby Serdio([28,Theorem10℄)in theparti ular asein whi h

g

isa onstant. Lemma7. Itholds:

N (f ∗ g) = N (f ) N (g).

Proof. Let

g

bexed. Wemustprovethat thetworealquadrati forms

Q

1

(f ) := (f ∗ g) ∗ (¯

g ∗ ¯

f )

and

Q

2

(f ) := (f ∗ ¯

f )(g ∗ ¯

g)

oin ide.Bypolarization,theequalityof

Q

1

and

Q

2

isequivalenttotheequality ofthe

R

bilinearsymmetri forms

B

1

(f

1

, f

2

) := (f

1

∗ g) ∗ (¯

g ∗ ¯

f

2

) + (f

2

∗ g) ∗ (¯

g ∗ ¯

f

1

),

B

2

(f

1

, f

2

) := (f

1

∗ ¯

f

2

)N (g) + (f

2

∗ ¯

f

1

)N (g).

By linearity, it is su ient to prove that

B

1

and

B

2

oin ide for

f

1

= w

i

_a

,

f

2

= w

j

b

, with

a, b ∈ O

. Sin e

B

α

(w

i

_{a, w}

j

_{b) = w}

i+j

_{∗ B}

α

(a, b)

(

α = 1, 2

), it

totheequality

Q

1

(a) = Q

2

(a)

forevery

a ∈ O

. Nowwe anx

a ∈ O

andapply thesameargumenttothequadrati forms

Q

′

1

(g) := (a ∗ g) ∗ (¯

g ∗ ¯

a)

and

Q

′

2

(g) := |a|

2

(g ∗ ¯

g).

We get that these two forms oin ide if and only if

Q

′

1

(b) = Q

′

2

(b)

for every

b ∈ O

. Thereforeitsu estoprovetheequality

(a b)(¯b ¯

a) = |a|

2

_|b|

2

forevery onstant

a, b ∈ O

.

Ineveryalternativealgebrathesubalgebragenerated bytwoelementsis asso- iative. Therefore

(a b)(¯b ¯

a) = a(b¯b)¯

a = |a|

2

_|b|

2

. This ompletestheproof. Corollary8. Itholds:

[

α∈V (f ∗g)

S

α

=

[

α∈V (f )∪V (g)

S

α

or, equivalently, given any

α ∈ O

,

V (f ∗ g) ∩ S

α

is nonempty if and only if

(V (f ) ∪ V (g))∩S

α

isnonempty. Inparti ular, thezerosetof

f ∗ g

is ontained

inthe unionof spheres

S

α∈V (f )∪V (g)

S

α

.

Proof. By ombiningCorollary5andLemma 7,weget:

S

α∈V (f ∗g)

S

α

= V (N (f ∗ g)) = V (N (f )) ∪ V (N (g)) =

=



S

α∈V (f )

S

α



∪



S

α∈V (g)

S

α



=

S

α∈V (f )∪V (g)

S

α

.

Sin e

V (f ∗ g) ⊂ V (N (f ∗ g))

, theproofis omplete.

Nowwegiveapre isedes riptionof thezerosoftheprodu t

f ∗ g

. Thanks to Corollary8, itis su ient to analyze separately thespheres

S

α

ontaining zerosof

f

orof

g

.

Firstly,we onsider thesigni ant ase: thenonrealzeros.

Let

α ∈ O

be nonreal. Supposethat

r

α

(f )(w) = wa + b

and

r

α

(g)(w) =

wc + d

forsome

a, b, c, d ∈ O

. Thenwehavethat

r

α

(f ∗ g)(w) = w(ad + bc + t

α

ac) + bd − n

α

ac = (r

α

(f ) ∗ r

α

(g))(w) − ∆

α

ac.

(3)

Thelatterassertionisanimmediate onsequen eofthefollowingfa t

(wa + b) ∗ (wc + d) =

w

2

(ac) + w(ad + bc) + bd =

=

(∆

α

+ wt

α

− n

α

)(ac) + w(ad + bc) + bd

ThankstoEq.(3),weobtain:

(1)

Suppose

f

has anisolatedzero

α

in

S

α

of multipli ity

s

and

g

isnowhere

zero on

S

α

. Then

f ∗ g

has an isolated zero

α

′

in

S

α

of multipli ity

s

,

ad + (¯

αa)c 6= 0

anditholds:

α

′

_{= ((αa)d + n}

α

ac) (ad + (¯

αa)c)

−1

.

Inparti ular, if

(αa)d = α(ad)

and

n

α

ac = α ((¯

αa)c) (

for example,if

a

is

realor if

a, b, c, d

belong toanasso iative subalgebraof

O )

,then

α

′

_{= α}

.

(2)

Suppose

f

is nowhere zero on

S

α

and

g

has an isolated zero

β

in

S

α

of multipli ity

t

. Then

f ∗ g

has an isolated zero

β

′

in

S

α

of multipli ity

t

,

bc + a( ¯

βc) 6= 0

anditholds:

β

′

_{= (b(βc) + n}

α

ac) bc + a( ¯

βc)

−1

.

Inparti ular, if

b(βc) = (bβ)c

and

a( ¯

βc) = (a ¯

β)c (

for example, if

c

isreal

orif

a, b, c, d

belongstoanasso iativesubalgebraof

O )

,thenwehavethat

β

′

_{= (b + a ¯}

_{β )β(b + a ¯}

_{β )}

−1

.

(3)

Suppose

f

has an isolated zero

α

in

S

α

of multipli ity

s

and

g

has an isolatedzero

β

in

S

α

ofmultipli ity

t

. Then,if

a(βc) = (¯

αa)c

,then

α

isa spheri alzeroof

f ∗ g

ofmultipli ity

s + t

. Otherwise,

f ∗ g

hasanisolated zero

γ

in

S

α

ofmultipli ity

s + t

su hthat

γ = (−(αa)(βc) + n

α

ac) (−a(βc) + (¯

αa)c)

−1

.

(4)

Let

α

beaspheri alzeroof

f

orof

g

,andlet

s, t

betherespe tive multipli -itiesof

α (

possiblyzero

)

. Then

α

isaspheri al zeroof

f ∗ g

ofmultipli ity

s + t

.

Proof. In thefollowing,we willuse thefa t that, in anyalternativequadrati algebra,thetra eisasso iativeand ommutative( f.e.g.[6,Ÿ9.1.2℄).

First,weprovethetheoremwithoutthepart on erningmultipli ities.

(1)Inthis ase

a 6= 0

and

αa + b = 0

( f.Corollary2(2)(ii)). If

c = 0

,then

d 6= 0

(otherwise

g

wouldvanish onthewhole sphere

S

α

). Then

ad + (¯

αa)c =

ad 6= 0

andEq.(3)givestheisolatedzero

α

′

_{= −(bd)(ad)}

−1

_{= ((αa)d)(ad)}

−1

of

f ∗ g

. If

c 6= 0

,then

ad + (¯

αa)c

annotvanish. Infa t,ifitwere

ad + (¯

αa)c = 0

,

then we would have that

α = −((ad)c

¯

−1

_)a

−1

. But then

tr(α) = tr(−dc

−1

₎

and

|α| = | − dc

−1

_|

. This would mean that

β = −dc

−1

belongsto

S

α

∩ V (g)

, ontradi tingourassumptions.

Corollary8tellsthat

S

α

must ontainazeroof

f ∗ g

. FromCorollary2(2)(ii) andEq.(3),thiszeromustbe

α

′

_{= (−bd + n}

α

ac)(ad + bc + t

α

ac)

−1

.

Note that the o tonion

ad + bc + t

α

ac

does not vanish, sin e it is equal to

ad − (αa)c + (t

α

a)c = ad − (αa)c + (αa)c + (¯

αa)c = ad + (¯

αa)c

. We on lude

that

α

′

_{= ((αa)d + n}

(2)This se ond ase issimilarto therstone. Now

c 6= 0

and

βc + d = 0

.

If

a = 0

,then

b 6= 0

and

bc + a( ¯

βc) = bc 6= 0

. FromEq.(3)wegettheisolated

zero

β

′

_{= (b(βc))(bc)}

−1

_.

If

a

doesnotvanish,thenitstill holds

bc + a( ¯

βc) 6= 0

. Infa t,ifthiswere nottrue,wewouldhavethat

β

is onjugateto

−ba

−1

and therefore

−ba

−1

_{∈ S}

α

∩ V (f )

. ByusingCorollary8,Corollary2(2)(ii),

βc = −d

and

t

α

= t

β

,weobtaintheuniquezero

β

′

_{= (−bd + n}

α

ac)(ad + bc + t

α

ac)

−1

= (b(βc) + n

α

ac)(bc + a( ¯

βc))

−1

of

f ∗ g

in

S

α

.

(3) In this ase,

a, c 6= 0

,

b = −αa

,

d = −βc

. From Eq. (3), we get the remainder:

r

α

(f ∗ g)(w) = w(−a(βc) − (αa)c + t

α

ac) + (αa)(βc) − n

α

ac

= w(−a(βc) + (¯

αa)c) + (αa)(βc) − n

α

ac.

If

a(βc) = (¯

αa)c

,then

r

α

(f ∗ g)

is onstant. Ontheotherhand,byCorollary8,

r

α

(f ∗ g)

vanishesonsomepointof

S

α

andhen eitisnull. Itfollowsthat

f ∗ g

has

α

asaspheri alzero. Instead,if

a(βc) 6= (¯

αa)c

,then

f ∗ g

hastheisolated zero

γ = (−(αa)(βc) + n

α

ac) (−a(βc) + (¯

αa)c)

−1

.

(4) If the real polynomial

∆

α

divides

f

or

g

, then it divides

f ∗ g

. The on lusionfollowsfromCorollary2(2)(i).

Now we onsider multipli ities. If

α

is a zero of

f

of multipli ity

s

and

β ∈ S

α

isa zeroof

g

ofmultipli ity

t

, then

N (f ) = ∆

s

α

h

1

and

N (g) = ∆

t

α

h

2

, where

h

1

and

h

2

real regular fun tions su h that

r

α

(h

1

) = wa + b 6= 0

and

r

α

(h

2

) = wc + d 6= 0

. FromLemma 7, itfollowsthat

N (f ∗ g) = ∆

s+t

α

(h

1

h

2

)

. Wehaveto provethat

∆

s+t+1

α

∤ N (f ∗ g)

. Byadensityargument,wesee that

this is equivalent to show that

∆

α

∤ h

1

h

2

, i.e. that theremainder

r

α

(h

1

h

2

)

is notzero. FromEq.(3),

r

α

(h

1

h

2

)

iszeroifandonlyif

ad + bc + t

α

ac = bd − n

α

ac = 0.

(4)

Sin e

a, b, c, d

are real, Eq. (4) impliesthat

a(d + αc)(d + ¯

αc) = 0

. But this annothappensin e

α

isnonrealand

a 6= 0

.

Example 1. Foreverynonreal

α ∈ O

,

α

¯

is onjugatedto

α

. Let

a

besu hthat

¯

α = aαa

−1

. If

f (w) = wa−αa

,

g(w) = w−α

,then

f ∗g = w

2

_{a−w(aα+αa)+αaα}

and

V (f ) = V (g) = {α}

while

V (f ∗ g) = S

α

(seeTheorem9(3)).

Remark 3. In the o tonioni ase, the amshaft ee t appears even if oneof the fun tions is onstant: the zeros of

f

and of

f ∗ c

(

c ∈ O

a onstant) an bedierent( f.Serdio[27℄when

f

isapolynomial). Forexample,let

f (w) =

wi − j

,

g(w) = k

. Then

f ∗ g = w(ik) − jk

has

−ij

as unique zero, while

V (f ) = {ij}

.

Proposition 10. Let

α

be a real o tonion. Suppose that

α

is a zero of

f

of multipli ity

s

and azeroof

g

of multipli ity

t ( s

and

t

possibly zero

)

. Then

α

isazeroof

f ∗ g

ofmultipli ity

s + t

.

Proof. If the real polynomial

w − α

divides

f

or

g

, then it divides

f ∗ g

. In parti ular,

r

α

(f ∗ g) = 0

. Moreover, if

f = (w − α)

s

_h

1

and

g = (w − α)

t

_h

2

with

h

1

(α) 6= 0

and

h

2

(α) 6= 0

, then

f ∗ g = (w − α)

s+t

_(h

1

∗ h

2

)

. It remains toprovethat

(w − α)

s+t+1

_{∤ f ∗ g}

. Bydensity, this isequivalenttoshowthat

w − α ∤ h

1

∗ h

2

, i.e.

(h

1

∗ h

2

)(α) 6= 0

. But

(h

1

∗ h

2

)(α) = 0

would imply

that

N (h

1

∗ h

2

)(α) = N (h

1

)(α) · N (h

2

)(α) = 0

. On the other hand, sin e

α

is real,

N (h

1

)(α) = |h

1

(α)|

2

_{6= 0}

and

N (h

2

)(α) = |h

2

(α)|

2

_{6= 0}

, whi h is a ontradi tion. 4 Appli ations

Webeginwithano tonioni versionofthefundamentaltheoremofalgebra. Theorem11 (Fundamental theorem of algebra). If

f

isa regularo tonioni polynomial of degree

n > 0

,then its zero set

V (f )

is nonempty and the ar-dinality

k

of the distin t onjuga y lasses of elements of

V (f )

is less than or equal to

n

. More pre isely, if

α

1

, . . . , α

r

are the distin t real zeros of

f

,

α

r+1

, . . . , α

r+i

are the distin t isolated zeros of

f

and

α

r+i+1

, . . . , α

r+i+s

are

pairwisenon onjugatespheri al zerosof

f

su hthat

S

s

j=1

S

α

r+i+j

isthe setof allspheri al zerosof

f

,then

k = r + i + s

andthe following equality holds:

k

X

j=1

m

f

(α

j

) = n.

Inparti ular, wehave that

r + i + 2s ≤ n

.

Proof. Thenormalpolynomial

N (f )

isarealpolynomialofdegree

2n

. Let

I ∈ S

.

Thentheset

V

I

(N (f )) := {z ∈ C

I

| N (f )(z) = 0} = C

I

∩

S

α∈V (f )

S

α

is non emptyand ontainsat most

2n

elements. Corollary5tellsthat

V (f ) ∩ S

α

6= ∅

forevery

α

su hthat

V

I

(N (f )) ∩ S

α

6= ∅

. Therefore,

V (f )

isnonempty.

If

f = ∆

α

h + wa + b

forsome regular fun tion

h

and

a, b ∈ O

, then

f =

¯

∆

α

¯

h+w¯

a+¯b

. Thisimpliesthatanisolatedzero

α = −ba

−1

of

f

,with

m

f

(α) = s

,

orrespondstoanisolatedzero

α = −¯b¯

ˆ

a

−1

_{∈ S}

α

of

f

¯

withthesamemultipli ity. From Theorem 9(3), we get that

α

is a spheri al zero of

N (f ) = f ∗ ¯

f

of multipli ity

2s

. The same property holds for spheri al and real zeros of

f

. Then,given

α

1

, . . . , α

k

as inthestatementofthetheorem,itfollowsthat

2

k

X

j=1

m

f

(α

j

) =

k

X

j=1

m

N

(f )

(α

j

) = 2n.

Sin e the multipli ity of a spheri al zero is at least

2

, it follows immediately

andofTheorem11gives:

Corollary12(Fa torizationlemma). Let

f

bearegularo tonioni polynomial of degree

n > 0

. Let

α

1

∈ V (f )

. Then thereexist

α

2

, . . . , α

n

, c ∈ O

with

c 6= 0

su hthat

f

fa tors asfollows:

f (w) = (w − α

1

) ∗ f

1

(w),

where

f

k

(w) : = (w − α

k+1

) ∗ f

k+1

(w)

for

k = 1, 2, . . . , n − 2

and

f

_n−1

(w) : = (w − α

n

) ∗ c.

Moreover, forevery

k = 1, . . . , n − 1

,

α

k+1

isazeroof

f

k

anditis onjugateto azero

α

′

k+1

of

f

.

Remark 4. Theorem9tellshowto obtainthezeros

α

′

k+1

∈ V (f )

from theset

{α

i

}

i=2,...,n

.

A standardappli ation ofthefundamental theorem of algebrain the om-plexeldistheuniquenessofmoni polynomialswithpres ribedzeros. Onthe quaternionsandontheo tonions,anewphenomenonappears. The multipli i-ty of an isolated zero of the sumof two regular fun tions an bestri tly less thanthoseofthefun tions. This fa timpliesthat twodierentmoni polyno-mials an havethesame zeroswith thesame multipli ities. Forexample, the polynomials

f (w) = (w − i) ∗ (w − i)

and

g(w) = (w − i) ∗ (w − j)

have the uniqueisolatedzero

w = i

withmultipli ity2,whilethedieren e

f − g

hasa simplezero

i

. Thisfa t annothappenwhenallthezerosarerealorspheri al orsimple,sin ein this ase

m

f+g

(α) ≥ min{m

f

(α), m

g

(α)}.

On the quaternions, the existen e of a unique moni regular polynomial withassignedpairwisenon onjugatezeroswasprovedbyBe k[1℄(seealso[3℄, [2℄ and [16℄). On the o tonions, it wasre ently proved by Serdio [28℄. The existen eofaninnitenumbersofmoni regularquaternioni polynomialswith pres ribedmultipleisolatedzeroswasprovedbyBe k[1℄.

An immediate onsequen e of the fundamental theorem and Propositions 3, 4 is the following result, whi h generalizes the one proved by Lam in the quaternioni ase( f.[22, Ÿ16.19℄):

Corollary 13. Let

f (w) =

P

n

i=0

w

i

a

i

be a regular o tonioni polynomial of degree

n > 0

. Thefollowing statementshold.

(1)

If the oe ients

a

1

, . . . , a

n

are real and

a

0

is nonreal, then

f

has only isolatedzeros.

(2)

If

a

2

, . . . , a

n

arereal andeither

a

0

or

a

1

isnonreal,then

f

has no

spher-i al zeros.

Inparti ular, theset

V (f )

ontainsexa tly

n

elements, ountedwiththeir mul-tipli ities.

onsider the polynomial

f (w) = w

2

_{+ wi + j}

. Sin e

N (f ) = w

4

_{+ w}

2

_{+ 1}

,

S

α∈V (f )

S

α

is equalto

S

β

∪ S

γ

, where

β :=

1

2

+ i

√

3

2

and

γ := −

1

2

+ i

√

3

2

. The orrespondingremaindersare

r

β

(f ) = w(1 + i) − (1 − j)

and

r

γ

(f ) = w(−1 + i) − (1 − j),

whi hgivethefollowingtwosimpleisolatedzerosof

f

:

α

1

=

1

2

(1−i−j−ij) ∈ S

β

and

α

2

=

1

2

(−1 − i + j − ij) ∈ S

γ

. Referen es

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