wu’s equations and quasi-hypergeometric functionswu’s equations and quasi-hypergeometric...

Commun. Math. Phys. 223, 475 – 507 (2001) Communications inMathematical

Physics© Springer-Verlag 2001

Wu’s Equations and Quasi-Hypergeometric Functions

Kazuhiko Aomoto1, Kazumoto Iguchi2

1 Graduate School of Mathematics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan.E-mail: [email protected]

2 70-3 Shinhari, Hari, Anan, Tokushima 774-0003, Japan. E-mail: [email protected]

Received: 22 October 1999 / Accepted: 28 June 2001

Abstract: We investigate analytic solutions to Wu’s equations and their symmetry prop-erty, singularities and monodromy, relating them to quasi hypergeometric functions. Wefinally give the monodromy theorem for them in one, two and arbitrary dimensionalcases (see Theorem 1, 2 and 3).

1. Quasi-Hypergeometric Functions Associated with Wu’s Equations

Let β ′ = (β ′i,j )ni,j=1 be ann × n matrix with real entriesβ ′i,j . Being given a point

z = (z1, . . . , zn) ∈ Cn, Wu’s equations with respect tow1, . . . , wn are described as

wi − 1= ziwβ′1i

1 · · ·wβ ′nin , (1.1)

1≤ i ≤ n.Wu’s equations have appeared as fundamental equations determining mutually frac-

tional exclusion statistics in statistical mechanics (see [25].) By applying the multivari-able Lagrange inversion formula, the second author has shown that the unique holomor-phic solution to (1.1) can be explicitly expressed as a special type of quasi hypergeometricfunctions (see [15]).

It is important to study the singularity of this solution both in mathematics and intheoretical physics, because it gives a global nature of the function and also is relatedto critical phenomena in statistical physics. However we do not discuss any physicalaspect.

The purpose of this note is to give a monodromy formula for the solutionsw(z)

to (1.1). Under a certain non-degenerate condition, we first give 3n kinds of the localsolutions to (1.1) near the pointsw = (w1, . . . , wn) such thatwj = 0,1,∞ in thecompactified space(CP 1)n.

476 K. Aomoto, K. Iguchi

To do that, we consider then dimensional real variety in the complex affine spaceCn,

XC1,... ,Cn :

arg w1−1

wβ′1,11 ···wβ

′n,1n

= C1

· · ·arg wn−1

wβ′1,n1 ···wβ

′n,nn

= Cn,(1.2)

depending on certain real constantsC1, · · · , Cn. The analytic continuation ofw(z) canbe done, in a concrete way, along special paths inXC1,... ,Cn .

The connection among the local solutions can be expressed as some congruenceidentities among the system ofn integers(ν1, . . . , νn).

Because of the symmetry of Wu’s equations, these functions also have a symmetry.At least in the case of a single variable case, this symmetry has already been observedin [23, 11, 14], etc.

We show that this symmetry plays a crucial part to get their global property likeanalytic continuation, singularity and monodromy.

The general theory on quasi-hypergeometric functions has been investigated in ourprevious paper (see [3, 4]). See also [9] where the authors investigate the very generalcategory of functions called “GG functions” which contain our quasi-hypergeometricfunctions as a special case.

We denote byfi(w1, . . . , wn) the functions

fi(w) = wi − 1− ziw1β ′1i · · ·wnβ ′ni . (1.3)

Then the JacobianJ = ∂(f1,... ,fn)∂(w1,... ,wn)

at a pointw = (w1, . . . , wn) where the equalitieshold

f1(w) = f2(w) = · · · = fn(w) = 0

is given by the formula

J = ϕ(w1, . . . , wn)

w1 · · ·wn .

Here ϕ(w1, . . . , wn) denotes the determinant of thenth order matrix(δi,jwj −β ′i,j (wj − 1))ni,j=1.

ϕ(w1, . . . , wn) is a polynomial ofnth degree which can be expressed as

ϕ(w) = B +n∑r=1

∑1≤i1<···<ir≤n

Ci1···ir wi1 · · ·wir , (1.4)

whereB andG( = C1,2,... ,n) denote the determinant of the matricesβ ′ and 1− β ′ re-spectively.

Let α1, . . . , αn be arbitrary complex numbers. We denote byF(z) = Fβ ′(α1, . . . , αn; z1, . . . , zn) the holomorphic function ofz at the origin (we call it the“Quasi-Hypergeometric Function”) which is by definition expressed as

Wu’s Equations and Quasi-Hypergeometric Functions 477

F(z) = Fβ ′(α1, . . . , αn; z1, . . . , zn)

=∑

ν1,... ,νn≥0

∏ni=1�(αi +

∑nk=1 β

′i,kνk)∏n

i=1�(αi +∑nk=1 βi,kνk)

zν11 · · · zνnnν1! · · · νn! (1.5)

for βi,i = β ′i,i − 1 , βi,k = β ′i,k (i �= k).As was proved in [15] (see also [1]) the following identity holds:

F(z) =[wα11 · · ·wαnnϕ(w)

]f1=···=fn=0

. (1.6)

The monomialwα11 · · ·wαnn itself has the power series

wα11 · · ·wαnn = KFβ ′(α1, . . . , αn; z1, . . . , zn), (1.7)

where

KFβ ′(α1, . . . , αn; z1, . . . , zn) = BFβ ′(α1, . . . , αn; z1, . . . , zn)

+n∑r=1

∑1≤i1<···<ir≤n

Ci1,... ,ir Fβ ′(α1, . . . , αi1 + 1, . . . , αir + 1, . . . , αn; z). (1.8)

Let us investigate the symmetry property of Eqs. (1.1). We define first a finite numberof transformationsβ

′ = (β ′i,j )ni,j=1: ρkβ ′, (1 ≤ k ≤ n − 1), σkβ ′ (1 ≤ k ≤ n) and

τkβ′ (1 ≤ k ≤ n) respectively applied toβ ′. Let ρk be the transposition between

the argumentsk and k + 1: β ′i,j = βρk(i),ρk(j). σkβ′ is defined by the substitution

β ′k,k = 1−β ′k,k, β ′k,j = −β ′k,j (j �= k), andβ ′i,j = β ′i,j (i �= k). Finallyτkβ ′ is defined as

β ′k,k = 1β ′k,k, β ′k,j =

β ′k,jβ ′k,k

(j �= k), β ′i,j =β ′k,kβ ′i,j−β ′k,j β ′i,k

β ′k,k(i, j �= k), β ′i,k = − β ′i,k

β ′k,k(i �= k).

We haveρ2i = σ 2

j = τ2k = e for the identical transformatione.

The system of generators{ρk(1 ≤ k ≤ n − 1), σk, τk(1 ≤ k ≤ n)} forms a finitegroupG of order 6n · n!. G is isomorphic to the semi-direct product ofn pieces ofS3andSn,

G ∼= Sn3 � Sn,whereSm denotes the symmetric group ofmth degree. {ρ1, . . . , ρn−1} with the relationsρiρi+1ρi = ρi+1ρiρi+1 generateSn. Each {σi, τi} generates the subgroupS3 with therelationτiσiτi = σiτiσi . σi, τi commute withσj , τj for i �= j .

We denote byG0 the subgroup of order 2n·n!generated by {ρ1, . . . , ρn−1, σ1, . . . , σn}.The right cosetG0\G is finite and has the cardinality|G0\G| = 3n. In particular we have

τ1 · · · τnβ ′ = β ′−1,

τ1 · · · τnσ1 · · · σnβ ′ = (1− β ′)−1.

We denote byβ ′(i1, . . . , ipj1, . . . , jp

)the subdeterminant of the(i1, . . . , ip)th lines and the

(j1, . . . , jp)th columns ofβ ′ and abbreviate it byβ ′(i1, . . . , ip) if i1 = j1, . . . , ip =

jp. We assume now the following non-degeneracy condition forβ ′:


σi1 · · · σipβ ′(j1, . . . , jq) �= 0, (ND)

1≤ i1 < · · · < ip ≤ n(0 ≤ p ≤ n),1≤ j1 < · · · < jq ≤ n(1≤ q ≤ n).In particular we have

BGβ ′1,1 · · ·β ′n,n(β ′1,1 − 1) · · · (β ′n,n − 1) �= 0.

The condition (ND) is invariant under the action ofG.

Lemma 1. Under (ND) Eqs. (1.1)admit the action of G as follows: We have

wi − 1= zi wβ′1,i

1 · · · wβ′n,in 1≤ i ≤ n, (1.9)

ρk : wi = wi(i �= k, k + 1), wk+1 = wk, wk = wk+1,

zi = zi(i �= k, k + 1), zk+1 = zk, zk = zk+1,

σk : wi = wi (i �= k), wk = w−1k ,

τk : wi = wi(i �= k), wk = 1− wk = −zkwβ′1,k

1 · · ·wβ′n,kn ,

zi = zi(−zk)− β′

kiβ′k,k (i �= k), zk = −(−zk)

− 1β′k,k .

Each representativeσ of the right cosetG0\G gives rise to the transformationF =Fβ ′(α; z)→ TσF satisfying the identity

Tσσ ′F = Tσ (Tσ ′F)for arbitraryσ, σ ′ ∈ G, such that

σ{wα11 · · ·wαnn

} = KTσFβ ′(α1, . . . , αn; z1, . . . , zn) σ ∈ G. (1.10)

In fact,

TρkFβ ′(α; z) = Fρkβ ′(α1, . . . , αk+1, αk, . . . , αn; z1, . . . , zk+1, zk, . . . , zn), (1.11)

TσkFβ ′(α; z) = Fσkβ ′(α1, . . . ,1− αk, . . . , αn; z1, . . . ,−zk, . . . , zn), (1.12)

TτkFβ ′(α; z) =1

β ′k,k(−zk)

− αkβ′k,k Fβ ′(α; z), (1.13)

whereαi = αi − αkβ′i,k

β ′k,k, αk = αk

β ′k,k, zk = −(−zk)

− 1β′k,k , zi = zi(−zk)

− β′k,i

β′k,k .

In particular

Tτ1···τnβ ′Fβ ′(α; z) =1

B(−z1)−α1 · · · (−zn)−αnFτ1···τnβ ′(α; z), (1.14)

whereαi =∑nj=1 β

′i,j αj for β

′ = β ′−1 andzi = −(−z1)− ˜β ′1,i · · · (−zn)− ˜β ′n,i .Similarly we have

Tσ1···σnτ1···τnFβ ′(α; z) =1

G(z1)

−α1 · · · (zn)−αnFτ1···τnσ1···σnβ ′(α; z) (1.15)

for αi =∑nj=1

˜β ′i,j αj for β′ = (1− β ′)−1 andzi = −(z1)− ˜β ′1,i · · · (zn)− ˜β ′n,i .


2. A Family of Local Solutions to Wu’s Equations

Consider Eqs. (1.1) in the compactified space(CP 1)n. We fix non negative integersp, qsuch thatp + q ≤ n. Put

w1 = Z1W1, ...., wp = ZpWp,wp+1 = Z−1p+1Wp+1, . . . , wp+q

= Z−1p+qWp+q, wp+q+1 = Wp+q+1, . . . , wn = Wn.

Then (1.1) are transformed into

ZiWi − 1 = −Wβ ′1,i1 · · ·Wβ ′n,i

n , 1≤ i ≤ p,Wi − Zi = Wβ ′1,i

1 · · ·Wβ ′n,in , p + 1≤ i ≤ p + q, (2.1)

Wi − 1 = ZiWβ ′1,i1 · · ·Wβ ′n,i

n , p + q + 1≤ i ≤ n,respectively, whereZ1, . . . , Zn are uniquely defined by the equations

ziZβ ′1i1 · · ·Zβ

′p,ip Z

−β ′p+1,ip+1 · · ·Z−β

′p+q,i

p+q = −1, 1≤ i ≤ p,ziZ

β ′1i1 · · ·Zβ

′p,ip Z

−β ′p+1,ip+1 · · ·Z1−β ′i,i

i · · ·Z−β′p+q,i

p+q = 1, p + 1≤ i ≤ p + q,ziZ

β ′1i1 · · ·Zβ

′p,ip Z

−β ′p+1,ip+1 · · ·Z−β

′p+q,i

p+q = Zi, p + q + 1≤ i ≤ n.Under the condition (ND), (2.1) has the power series solutions inZ1, . . . , Zn in

a small neighborhood of the origin|Z1| < δ1, . . . , |Zn| < δn for suitable positivenumbersδ1, . . . , δn,

Wi = 1+n∑k=1

Zkck,i + · · · . (2.2)

ck,i are uniquely determined by the relations

n∑j=1

ci,j β′j,s = −δi,s 1≤ i ≤ p, 1≤ s ≤ n,

n∑j=1

ci,j β′j,s − ci,s = −δi,s , p + 1≤ i ≤ p + q, 1≤ s ≤ n,

ci,s = δi,s , p + q + 1≤ i ≤ n, 1≤ s ≤ n.We can choose an arbitrary subset of indicesi1, . . . , ip, ip+1, . . . , ip+q from{1,2, . . . , n} instead of the subset{1,2, . . . , p;p + 1, . . . , p + q}. We have thena similar solution to (2.2).

We denote by〈ε1, . . . , εn〉 the local solution to (1.1) such thatεj = 0,∞,1 aswjtakes 0,∞,1 atZ1 = Z2 = ... = Zn = 0, in other words, asj = iν (1 ≤ ν ≤ p),j = iν (p + 1 ≤ ν ≤ p + q) andj otherwise. The total number of the local solutionsthus obtained is 3n. In particular, whenq = n, we have the local solution〈0n〉 such that

wi = Zi(

1−n∑j=1

β ′i,jZj + · · ·)

(1≤ i ≤ n), (2.3)


where

Zi = (−z1)−β ′1,i · · · (−zn)−β ′n,i 1≤ i ≤ n, (2.4)

and the matrix(β ′i,j

)denotes the inverse ofβ ′.

Definition 1. In the complex plane zj near the infinity, 〈0n〉 is multivalued and has newbranches by counterclockwise rotations Sj zj → zj e

2πi . We denote the branches thusobtained by

w(∞)ν1,... ,νn= Sν1

1 · · · Sνnn 〈0n〉 = exp

(−2πi

n∑k=1

β ′k,iνk)Zi(1− · · · ). (2.5)

3. CaseN = 1

Let us consider first the casen = 1 which plays a basic role in many variable cases.Let α ∈ C andβ ′ ∈ R be given constants. Equation (1.1) reduces to

w − 1= zwβ ′ (3.1)

and the QHGF defined in (1.4) reduces to the Lambert series

Fβ ′(α; z) =∞∑ν=0

�(α + β ′ν)�(α + (β ′ − 1)ν)

zν

ν! . (3.2)

We have the identities

Fβ ′(α; z) = wα

(1− β ′)w + β ′ , (3.3)

wα = β ′Fβ ′(α; z)+ (1− β ′)Fβ ′(α + 1; z) (3.4)

for the holomorphic solutionw to (1.1) at the origin. In particular,

w = β ′Fβ ′(1; z)+ (1− β ′)Fβ ′(2; z). (3.5)

We shall denote this holomorphic solutionw byw(0)0 . The condition (ND) simply meansβ ′ �= 0,1. Equation (3.1) has the symmetry with respect to

σ1 : β ′ → 1− β ′

and

τ1 : β ′ → 1

β ′.

In the sequel we shall only consider the case β ′ > 1,without losing generality.Fβ ′(α; z)is holomorphic in the disc|z| < c (c = β ′−β ′(β ′ − 1)β

′−1) and has a branch atz = c.We have

Tσ1Fβ ′(α; z) = F1−β ′(1− α;−z) = Fβ ′(α; z) (3.6)


for |z| < c. On the other hand,

Tτ1Fβ ′(α; z) =1

β ′(−z)− α

β′ F 1β′

(α

β ′; −(−z)− 1

β′)

(3.7)

which is defined in a neighborhood ofz = ∞. In the same way

Tτ1σ1Fβ ′(α; z) =1

β ′(−z)− α

β′ F1− 1β′

(1− α

β ′; (−z)− 1

β′), (3.8)

Tσ1τ1Fβ ′(α; z) =1

1− β ′ z− 1−α

1−β′ F 11−β′

(1− α1− β ′ ; −z

1β′−1

), (3.9)

Tτ1σ1τ1Fβ ′(α; z) =1

1− β ′ z− 1−α

1−β′ F β′β′−1

(α − β ′1− β ′ ; −z

1β′−1

). (3.10)

The map

T : w→ w − 1

wβ′ (3.11)

from R>0 to R has critical pointw(c) = β ′β ′−1 > 1 and critical valuec.

The inverse image byT of the interval(0,∞) consists of the real positive linel1 andthe convex loopl2 starting from and ending in the origin, which intersectsl1 atw(c) innormal crossing. The arguments of its tangents at the origin are equal to± π

β ′ . l1 andl2

form the separatrices of the phase defined by the family of curves

XC : argw − 1

wβ′ = C

(C denotes a constant).

Lemma 2. Equation (3.1)has the local solution w(0)0 at z = 0 besides w(0)0 ,

w(0)0 = Z−1

(1− 1

β ′ − 1Z + · · ·

), (3.12)

which is a Laurent series with respect to Z for Z = z 1β′−1 . w(0)0 can also be expressed

by using (3.9)as

w(0)0 = β ′Tσ1τ1Fβ ′(1; z)+ (1− β ′)Tσ1τ1Fβ ′(2; z) (3.13)

from the identity (3.4).

Proof. By substitutionw = Z−1W ,Z = z1

β′−1 , w(0)0 can be obtained as the uniquesolution to the equation

W − Z = Wβ ′

such thatW(0) = 1.


In the same way, we have

Lemma 3. Equation (3.1) has the local solution w(∞)0 in a neighborhood of z = ∞,

w(∞)0 = Z

(1− 1

β ′Z + · · ·

)(3.14)

for Z = (−z)− 1β′ , and

w(∞)0 = β ′Tτ1Fβ ′(1; z)+ (1− β ′)Tτ1Fβ ′(2; z). (3.15)

Lemma 4.Tσ1τ1Fβ ′(α; z) (or w(0)0 ) is the analytic continuation of Fβ ′(α; z) (or w(0)0 )along a closed curve starting from the origin, turning around c counterclockwise in asmall circle and going back to the origin.

Let S be the shift operator defined by the analytic continuation by the rotationz→ze2πi , and hence bySν theν-times rotations for an integerν. We denoteSνw(0)0 , Sνw(∞)0

by w(0)ν , w(∞)ν respectively.

Lemma 5. Let l+ be the oriented curve consisting of the interval [1, w(c)] combinedby the lower half of l2 which starts from 1 and ends in w = 0. Then Tτ1Fβ ′(α; z) (or

w(∞)0 ) is the analytic continuation of Fβ ′(α; z) (or w(0)0 ) when z moves from 0 to +∞

along the real axis z ≥ 0 detoured around c in the lower half plane.

Lemma 6. Let l− be the oriented curve consisting of the interval [w(c),∞] and theupper half of l2 which starts from w = +∞ and ends in w = 0. Then w(∞)−1 is the

analytic continuation of w(0)0 along l− when z moves starting from z = 0 and ending inz = +∞ along the real axis detoured around c in the lower half plane.

We haveX0 = l1∪ l2 = l+ ∪ l−. HenceX0 contains two paths from 1 to 0 and from∞ to 0 along which we have the analytic continuations fromw(0)0 to w(∞)0 and from

w(0)0 tow(∞)−1 respectively.We also consider the following equations which are obtained from (3.1) by the rotation

w→ we2πνi (ν ∈ Z):

w − 1= e2πνiβ ′wβ′. (3.16)

Equation (3.16) has the holomorphic solutionw(0)ν at z = 0,

w(0)ν = β ′Fβ ′(1; e2πνβ ′iz)+ (1− β ′)Fβ ′(2; e2πνβ ′iz). (3.17)

Proposition 1. (1) Suppose that w(∞)µ is the local solution at z = ∞ obtained by the

analytic continuation ofw(0)ν along the real curveX0 from the origin z = 0 to z = +∞.Let 2πνβ ′ and −π+2πµ

β ′ be expressed as

2πνβ ′ = ϕ + 2mπ (3.18)


(0< ϕ ≤ 2π, m ∈ Z) and

−π + 2πµ

β ′= θ + 2nπ (3.19)

(|θ | < π, n ∈ Z.) Then we get the equalities

1+ µ = −m, ν = n. (3.20)

(2) Put

2πν

1− β ′ = ϕ + 2mπ (3.21)

for 0 ≤ ϕ < 2π .w(∞)µ is the analytic continuation of w(0)ν along the curve l2 if and onlyif

µ+ 1= ν −m. (3.22)

Proof. (1) The curveX0 lies in the region surrounded byl2. We have

C = ϕ = π − β ′θ. (3.23)

This implies (3.20).(2) In this case the curveX0 lies outside of the curvel2. We have

C = ϕ − β ′(ϕ + 2mπ) = π − β ′(θ + 2nπ). (3.24)

This occurs if and only if (3.22) holds.

This proposition shows that the local solutionsw(0)ν or w(0)ν at the origin are analyti-cally continued to the onesw(∞)µ at z = ∞ along the curvesX0.

Lemma 7. As z moves from 0 to c on the real axis and turn around c, in a detouredway and moves back to the origin, w(0)0 and w(0)0 meet each other and are transposed at

z = c, while each of any other branches w(0)µ or w(0)µ remains the same.

In the same way w(∞)0 and w(∞)−1 meet each other and are transposed while each of

the other w(∞)µ (µ �= 0,−1) remains the same.In other words, we have the monodromy which is a permutation among{

w(∞)µ

}−∞<µ<∞

Mc :

w(∞)0 → w

(∞)−1 ,

w(∞)−1 → w

(∞)0 ,

w(∞)µ → w

(∞)µ (µ �= 0,−1).

On the other hand by the rotation S, we have

S : w(∞)µ → w(∞)µ+1. (3.25)

As a conclusion, we may state

Theorem 1.Assume that β ′ > 1. The monodromy corresponding to the group generatedby the transformations Mc and S gives rise to permutations among the local solutionsw(∞)µ (−∞ < µ < ∞). It contains every finite permutation and the shifts Sν (−∞ <

ν <∞):

Sν : w(∞)µ → w(∞)µ+ν . (3.26)


4. Casen = 2

Equations (1.1) are written as

w1 − 1= z1wβ′1,1

1 wβ ′2,12 ,

w2 − 1= z2wβ′1,2

1 wβ ′2,22 . (4.1)

The corresponding QHGF is

Fβ ′(α1, α2; z1, z2)

= ∑ν1,ν2≥0

�(α1 + β ′1,1ν1 + β ′1,2ν2)�(α2 + β ′2,1ν1 + β ′2,2ν2)

�(α1 + β1,1ν1 + β1,2ν2)�(α2 + β2,1ν1 + β2,2ν2)

zν11 z

ν22

ν1!ν2! (4.2)

for β ′1,1 − 1 = β1,1 β′2,2 − 1 = β2,2 andβ ′1,2 = β1,2β

′2,1 = β2,1. Equations (1.3) and

(1.5) are valid for

ϕ(w1, w2) = Gw1w2 − (B − β ′1,1)w2 − (B − β ′2,2)w1 + B, (4.3)

or equivalently,

wα11 w

α22 = KFβ ′(α1, α2; z1, z2), (4.4)

whereK denotes the operator

KFβ ′(α1, α2; z1, z2)= GFβ ′(α1 + 1, α2 + 1; z1, z2)− (B − β ′1,1)Fβ ′(α1, α2 + 1; z1, z2)−(B − β ′2,2)Fβ ′(α1 + 1, α2; z1, z2)+ BFβ ′(α1, α2; z1, z2). (4.5)

HereB andG represent the determinants

B = β ′1,1β ′2,2 − β ′1,2β ′2,1,G = (1− β ′1,1)(1− β ′2,2)− β ′1,2β ′2,1.

In particular

w1 = KFβ ′(1,0; z1, z2),w2 = KFβ ′(0,1; z1, z2). (4.6)

The condition (ND) reduces to the inequality

BGβ ′1,1β ′2,2(β ′1,1 − 1)(β ′2,2 − 1)(B − β ′1,1)(B − β ′2,2) �= 0. (4.7)

The groupG is of order 72 and is generated by< ρ1, σ1, σ2, τ1, τ2 > with re-lationsρ2

1 = σ 21 = σ 2

2 = τ21 = τ2

2 = e, σ1σ2 = σ2σ1, τ1τ2 = τ2τ1, σ1τ1σ1 =τ1σ1τ1, σ2τ2σ2 = τ2σ2τ2. The representatives of the cosetG0\G can be chosen as{e, τ1, τ2, τ1τ2, τ1σ1, τ2σ2, τ1σ1τ2 , τ2σ2τ1 , τ1τ2σ1σ2} (see [13]).


The corresponding matrices are given as

σ1β′ =

(1− β ′1,1,−β ′1,2

β ′2,1, β ′2,2

), σ2β

′ =(

β ′1,1, β ′1,2−β ′2,1,1− β ′2,2

),

τ1β′ =

1

β ′1,1,β ′1,2β ′1,1

−β′2,1

β ′1,1, Bβ ′1,1

, τ2β

′ =

B

β ′2,2,−β ′1,2

β ′2,2β ′2,1β ′2,2

, 1β ′2,2

,

τ1τ2β′ =

β ′2,2B,−β

′1,2

B

−β′2,1

B,β ′1,1B

,

τ1σ1β′ =

1

1− β ′1,1,− β ′1,2

1− β ′1,1− β ′2,1

1− β ′1,1,β ′2,2 − B1− β ′1,1

, τ2σ2β

′ =

β ′1,1 − B1− β ′2,2

,− β ′1,21− β ′2,2

− β ′2,11− β ′2,2

,1

1− β ′2,2

,

τ2σ2τ1β′ =

1− β ′2,2β ′1,1 − B

,− β ′1,2β ′1,1 − B

β ′2,1β ′1,1 − B

,β ′1,1

β ′1,1 − B

, τ1σ1τ2β

′ =

β ′2,2β ′2,2 − B

,β ′1,2

β ′2,2 − B− β ′2,1β ′2,2 − B

,1− β ′1,1β ′2,2 − B

,

τ1τ2σ1σ2β′ =

1− β ′2,2G

,β ′1,2G

β ′2,1G,

1− β ′1,1G

.

Lemma 8. An arbitrary matrix β ′ such that β ′1,2β ′2,1 �= 0 is transformed by a suitable

element of G into a matrix β′

such that β ′1,2 > 0, β ′2,1 > 0.

Lemma 9. Every matrix β ′ satisfyingβ ′1,2 > 0, β ′2,1 > 0, belongs to one of the following4 orbits of G, according to the signs ofG,B, β ′1,1, β ′2,2, β ′1,1−1, β ′2,2−1, B−β ′1,1, B−β ′2,2:

(a) (++++++++) (b) (+−++−−−−)(++++−−−−) (−++++++−)(−++++−−+) (−+++++−+)(−+++−++−) (−+++++++)(+−−+−−+−) (−−−−−−−−)(+−+−−−−+) (+−−+−−−−)(−−−+−+−+) (+−+−−−−−)(−−+−+−+−) (−−+−−−−−)(++−−−−++), (−−−+−−−−),


(c) (−−++++−−) (d) (−−++−−−−)(−−+++−−−) (−+++++−−)(−−++−+−−) (+−−−−−−−).(−+++−+−−)(−++++−−−)(−+++−−−−)(+−−−−−++)(+−−−−−−+)(+−−−−−+−),

As a consequence, we have

Corollary 1. An arbitrary matrix β ′ satisfying ND such that β ′1,2 > 0, β ′2,1 > 0 canbe transformed by an element of G into the following four cases:

(a) β ′1,2 > 0, β ′2,1 > 0, B > 0,G > 0, β ′1,1 > 1, β ′2,2 > 1, B−β ′1,1 > 0, B−β ′2,2 > 0;(b) β ′1,2 > 0, β ′2,1 > 0;B < 0,G > 0,1 > β ′1,1 > 0,1 > β ′2,2 > 0, B − β ′1,1 <

0, B − β ′2,2 < 0;(c) β ′1,2 > 0, β ′2,1 > 0, B < 0,G < 0, β ′1,1 > 1, β ′2,2 > 1, B−β ′1,1 < 0, B−β ′2,2 < 0;(d) β ′1,2 > 0, β ′2,1 > 0, B < 0,G < 0,1 > β ′1,1 > 0,1 > β ′2,2 > 0, B − β ′1,1 <

0, B − β ′2,2 < 0.

In this section, because of simplicity,we only consider the case (a) in the corollary,namely we assume the following:

(C1) β′1,2 > 0, β ′2,1 > 0, B > 0,G > 0, β ′1,1 > 1, β ′2,2 > 1, B−β ′1,1>0, B−β ′2,2>0.

The transformation formula forFβ ′(α1, α2; z1, z2) corresponding to each represen-tative ofG0\G is given as follows:

(1) Tσ1Fβ ′(α1, α2; z1, z2) = Fσ1β′(1− α1, α2;−z1, z2),

Tσ2Fβ ′(α1, α2; z1, z2) = Fσ2β′(α1,1− α2; z1,−z2).

Both coincide withFβ ′(α1, α2; z1, z2) itself.

(2) Tτ1Fβ ′(α1, α2; z1, z2) = Fτ1β ′(α1β ′1,1, α2 − α1

β ′2,1β ′1,1

; z1, z2)

1β ′1,1(−z1)

− α1β′1,1

for z1 = −(−z1)− 1β′1,1 , z2 = z2(−z1)

− β′1,2β′1,1 .

(3) Tτ2Fβ ′(α1, α2; z1, z2) = Fτ2β ′(α1 − α2

β ′1,2β ′2,2, α2β ′2,2

; z1, z2)

1β ′2,2(−z2)

− α2β′2,2

for z1 = z1(−z2)− β′2,1β′2,2 , z2 = −(−z2)

− 1β′2,2 .

(4) Tτ1τ2Fβ ′(α1, α2; z1, z2) = 1B(−z1)−α1(−z2)−α2Fτ1τ2β ′(α1, α2; z1, z2)

for α1 = α1β′2,2−α2β

′1,2

B, α2 = −α1β

′2,1+α2β

′1,1

Band z1 = −(−z1)−

β′2,2B (−z2)

β′2,1B ,

z2 = −(−z1)β′1,2B (−z2)−

β′1,1B .


(5) Tσ1τ1Fβ ′(α1, α2; z1, z2) = Fτ1σ1β′(

1−α11−β ′1,1 , α2 − (1−α1)β

′2,1

1−β ′1,1 ; z1, z2)

11−β ′1,1 z

− 1−α11−β′1,1

1

for z1 = −z− 1

1−β′1,11 , z2 = z

β′1,21−β′1,11 z2.

(6) Tσ2τ2Fβ ′(α1, α2; z1, z2) = Fτ2σ2β′(α1 − (1−α2)β

′1,2

1−β ′2,2 , 1−α21−β ′2,2 ; z1, z2

)1

1−β ′2,2 z− 1−α2

1−β′2,22

for z1 = z1zβ′2,1

1−β′2,22 , z2 = −z

− 11−β′2,2

2 .

(7) Tτ1σ2τ2Fβ ′(α1, α2; z1, z2) = Fτ2σ2τ1β′(α1, α2; z1, z2) 1

β ′1,1−B (−z1)−α1(z2)

−α2

for α1 = α1(1−β ′2,2)−(1−α2)β′1,2

β ′1,1−B , α2 = α1β′2,1+(1−α2)β

′1,1

β ′1,1−B and z1 = −(−z1)− 1−β′2,2β′1,1−B

(z2)− β′2,1β′1,1−B , z2 = −(−z1)

β′1,2β′1,1−B (z2)

− β′1,1β′1,1−B .

(8) Tτ2σ1τ1Fβ ′(α1, α2; z1, z2) = Fτ1σ1τ2β′(α1, α2; z1, z2) 1

β ′2,2−B (z1)−α1(−z2)−α2

for α1 = (1−α1)β′2,2+α2β

′1,2

β ′2,2−B , α2 = −(1−α1)β′2,1+α2(1−β ′1,1)β ′2,2−B and z1 = −(z1)

− β′2,2β′2,2−B

(−z2)β′2,1β′2,2−B , z2 = −(z1)

− β′1,2β′2,2−B (−z2)

− 1−β′1,1β′2,2−B .

(9) Tσ1σ2τ1τ2Fβ ′(α1, α2; z1, z2) = Fτ2τ1σ2σ1β′(α1, α2; z1, z2) 1

Gz−α11 z

−α22

for α1 = (1−α1)(1−β ′2,2)+(1−α2)β′1,2

G, α2 = (1−α1)β

′2,1+(1−α2)(1−β ′1,1)

Gand z1 = −z−

1−β′2,2G

1

z− β′2,1

G

2 , z2 = −z−β′1,2G

1 z− 1−β′1,1

G

2 .

Proposition 2. Corresponding to each of the coset G0\G there are 9 local solutionsw = (w1, w2) to Eq. (4.1). They are given as follows:

(1) w = (w1, w2) are holomorphic in a neighborhoodU1(δ1, δ2) : |z1| < δ1, |z2| < δ2for small positive numbers δ1, δ2.

w1 = KFβ ′(1,0; z1, z2) = 1+ z1 + · · · ,w2 = KFβ ′(0,1; z1, z2) = 1+ z2 + · · · . (4.8)

(2) w1 = KTτ1Fβ ′(1,0; z1, z2) = Z1

(1− 1

β ′1,1Z1 −

β ′2,1β ′1,1

Z2 + · · ·),

w2 = KTτ1Fβ ′(0,1; z1, z2) = 1+ Z2 + · · · (4.9)


for Z1 = (−z1)− 1β′1,1 , Z2 = z2(−z1)

− β′1,2β′1,1 in a neighborhood U2(δ1, δ2) : |Z1| <

δ1, |Z2| < δ2.

(3) w1 = KTτ2Fβ ′(1,0; z1, z2) = 1+ Z1 + · · · ,

w2 = KTτ2Fβ ′(0,1; z1, z2) = Z2

(1− β

′1,2

β ′2,2Z1 − 1

β ′2,2Z2 + · · ·

)

for Z1 = z1(−z2)− β′2,1β′2,2 , Z2 = (−z2)

− 1β′2,2 in a neighborhood U3(δ1, δ2) : |Z1| <

δ1, |Z2| < δ2.

(4) w1 = KTτ1τ2Fβ ′(1,0; z1, z2) = Z1

(1− β

′2,2

BZ1 +

β ′2,1BZ2 + · · ·

),

w2 = KTτ1τ2Fβ ′(0,1; z1, z2) = Z2

(1+ β

′1,2

B− β

′1,1

BZ2 + · · ·

)

for Z1 = (−z1)−β′2,2B (−z2)

β′2,1B , Z2 = (−z1)

β′1,2B , (−z2)−

β′1,1B in a neighborhood

U4(δ1, δ2) : |Z1| < δ1, |Z2| < δ2.

(5) w1 = KTσ1τ1Fβ ′(1,0; z1, z2) = Z−11

(1+ 1

1− β ′1,1Z1 +

β ′2,11− β ′1,1

Z2 + · · ·),

w2 = KTσ1τ1Fβ ′(0,1; z1, z2) = 1+ Z2 + · · · (4.10)

for Z1 = z1

β′1,1−1

1 , Z2 = zβ′1,2

1−β′1,11 z2 in a neighborhood U5(δ1, δ2) : |Z1| < δ1, |Z2| < δ2.

(6) w1 = KTσ2τ2Fβ ′(1,0; z1, z2) = 1+ Z1 + · · · ,

w2 = KTσ2τ2Fβ ′(0,1; z1, z2) = Z−12

(1+ β ′1,2

1− β ′2,2Z1 + 1

1− β ′2,2Z2 + · · ·

)(4.11)

for Z1 = z1zβ′2,1

1−β′2,22 , Z2 = z

1β′2,2−1

2 in a neighborhood U6(δ1, δ2) : |Z1| < δ1, |Z2| < δ2.

(7) w1 = KTτ1σ2τ2Fβ ′(1,0; z1, z2) = Z1

(1+ β ′2,2 − 1

β ′1,1 − BZ1 −

β ′2,1β ′1,1 − B

Z2 + · · ·),

w2 = KTτ1σ2τ2Fβ ′(0,1; z1, z2) = Z−12

(1− β ′1,2

β ′1,1 − BZ1 +

β ′1,1β ′1,1 − B

Z2 + · · ·)

(4.12)

for Z1 = (−z1)β′2,2−1

β′1,1−B z

− β′2,1β′1,1−B

2 , Z2 = (−z1)β′1,2β′1,1−B z

− β′1,1β′1,1−B

2 in a neighborhood

U7(δ1, δ2) : |Z1| < δ1, |Z2| < δ2.


(8) w1 = KTτ2σ1τ1Fβ ′(1,0; z1, z2) = Z−11

(1+ β ′2,2

β ′2,2 − BZ1 −

β ′1,2β ′2,2 − B

Z2 + · · ·),

w2 = KTτ2σ1τ1Fβ ′(0,1; z1, z2) = Z2

(1− β ′2,1

β ′2,2 − BZ1 +

β ′1,1 − 1

β ′2,2 − BZ2 + · · ·

)(4.13)

for Z1 = z

− β′2,2β′2,2−B

1 (−z2)β′2,1β′2,2−B , Z2 = z

− β′1,2β′2,2−B

1 (−z2)β′1,1−1

β′2,2−B in a neighborhood

U7(δ1, δ2) : |Z1| < δ1, |Z2| < δ2.

(9) w1 = KTσ1σ2τ1τ2Fβ ′(1,0; z1, z2) = Z−11

(1+ 1− β ′2,2

GZ1 +

β ′2,1GZ2 + · · ·

),

w2 = KTσ1σ2τ1τ2Fβ ′(0,1; z1, z2) = Z−12

(1+ β

′1,2

GZ1 +

1− β ′1,1G

Z2 + · · ·)(4.14)

for Z1 = z

β′2,2−1

G

1 z− β′2,1

G

2 , Z2 = z− β′1,2

G

1 z

β′1,1−1

G

2 in a neighborhood U9(δ1, δ2) : |Z1| <δ1, |Z2| < δ2.

Generally Eqs. (4.1) have local solutions at each pointwwherew1, w2 equal 0, 1 or∞such that its restriction to a family of special curves in the space (z1, z2) can be expressedasquasi Puiseux expansions, as follows: Let us fix positive constantsv, λ. Consider thefamily of curvesγλ (or γ ∗λ ) : z1 = t, z2 = vtλ (or z1 = −t−1, z2 = −vt−λ), wheret moves over the set 0< t ≤ δ, δ being a small positive constant. On each curveγλ(or γ ∗λ ) Eqs. (4.1) with respect tow1, w2 have a finite number of solutions which areexpressed as quasi Puiseux expansions,

w1 = u1,0tκ1,0 + u1,1t

κ1,1 + · · · ,w2 = u2,0t

κ2,0 + u2,1tκ2,1 + · · · , (4.15)

such that

κ1,0 < κ1,1 < · · · ,κ2,0 < κ2,1 < · · ·

andu1,0 > 0, u2,0 > 0.Let us call these solutions “admissible”. Then

Proposition 3. (a) (1), (5), (6) and (9) are all admissible solutions to (4.1) which areholomorphic in the neighborhood

|z1|− β′1,2β′1,1−1 |z2| < δ1, |z1||z2|

− β′2,1β′2,2−1

< δ2, |z1| < δ3, |z2| < δ4 (4.16)

for small positive numbers δ1, δ2, δ3, δ4, having an expression (4.15)on γλ.


(b) (4)is the only admissible solution to (4.1)which is holomorphic in the neighborhood

|z1|− β′1,2β′1,1 |z2| > 1

δ1, |z1||z

− β′2,1β′1,1

2 | > 1

δ2, |z1| > 1

δ3, |z2| > 1

δ4(4.17)

having an expression (4.15)on γ ∗λ .

Proof. (a)λ satisfies the inequalities

β ′1,2β ′1,1 − 1

< λ <β ′2,2 − 1

β ′2,1.

Once the exponents (κ1,0, κ2,0) are determined, (4.15) is uniquely expressed in arecursive way. There are only 4 such exponents satisfying (4.1) which correspond to (1),(5), (6), (9).

(b) In this case we have the inequalities

β ′1,2β ′1,1

< λ <β ′2,2β ′2,1

.

It can be proved similarly as above.

5. Critical Set, Branch Loci and Analytic Continuation

Equations (4.1) are equivalent to considering the mapT :

T :

z1 = w1 − 1

wβ ′1,11 w

β ′2,12

z2 = w2 − 1

wβ ′1,21 w

β ′2,22

from the source spaceR2>0 into the target spaceR2. T has the critical set8 defined by

the equation

8 : ϕ(w1, w2) = 0. (5.1)

8 is the hyperbola with two asymptotic linesw1 = a1, andw2 = a2, centered at

(a1, a2)=(B−β ′1,1G

,B−β ′2,2G

). It has two components81 = 8 ∩ {w1 < a1, w2 < a2},

82 = 8 ∩ {w1 > a1, w2 > a2}.8 is parametrized by

w1 − a1 =β ′2,1Gt,w2 − a2 =

β ′1,2Gt. (5.2)

It is known since H.Whitney that the local singularities of a 2 dimensional mappinggenerically consist of folds, cusps and nodes (see [17, 19, 23].) Consider the restrictionof T to8 (denoted byT ′). T has a fold at a point of8 whereT ′ is not singular. Thesingular points ofT ′ corresponding to cusps ofT (more precisely81,1-type Boardmansingularity) is given by the equations

dz1

z1≡ dz2

z2≡ 0 moddϕ. (5.3)


This means the following cubic equation int :

− c1

t − t1 +c2

t − t2 −1

t − t3 +1

t= 0, (5.4)

wheret1 = −B−β′1,1

β ′2,1, t2 = −β

′2,2−1

β ′2,1, t3 = − β ′1,2

B−β ′2,2 andc1 = β ′1,1β ′2,1, c2 = 1

β ′2,1.

The critical sets are illustrated in Figs. 1(a)–(d) according to the 4 cases (a)–(d) inCorollary 1 to Lemma 9.

From now on we assume the condition (C1), i.e., only consider the case (a).Equation (5.4) has at least one positive solution which we denote byt0. It has 3

different real solutions if and only if its discriminant is positive. This is also equivalentto positivity of the following polynomial: in β ′1,1, β ′1,2, β ′2,1, β ′2,2 appearing in theirreducible factor of the discriminant:

: = :0,0 +:1,1β′1,2β

′2,1 +:2,1β

′1,2

2β ′2,1 +:1,2β

′1,2β

′2,1

2 +:2,2β′1,2

2β ′2,1

2

+:3,2β′1,2

3β ′2,1

2 +:2,3β′1,2

2β ′2,1

3 +:4,2β′1,2

4β ′2,1

2 +:2,4β′1,2

2β ′2,1

4

+:3,3β′1,2

3β ′2,1

3 +:4,3β′1,2

4β ′2,1

3 +:3,4β′1,2

3β ′2,1

4 + β ′1,24β ′2,1

4,

where

:0,0 = − 27β ′1,12β ′1,1

2(β ′2,2 − 1)2(β ′2,2 − 1)2,

:1,1 = 18(β ′1,1 − 1)(β ′2,2 − 1)(2β ′1,1 − 1)(2β ′2,2 − 1),

:2,1 = 2β ′1,1(β ′1,1 − 1)(β ′2,2 + 1)(2β ′2,2 − 1)(β ′2,2 − 2),

:1,2 = 2β ′2,2(β ′2,2 − 1)(β ′1,1 + 1)(2β ′1,1 − 1)(β ′1,1 − 2),

:2,2 = − 62β ′1,1β ′2,2(1− β ′1,1)(1− β ′2,2)+ 8(β ′1,1 + β ′2,2 − β ′1,12 − β ′1,12)+ 1,

:3,2 = 2(2β ′2,22 − 2β ′2,2 − 1)(1− 2β ′1,1),

:2,3 = 2(2β ′1,12 − 2β ′1,1 − 1)(1− 2β ′2,2),

:4,2 = :2,4 = 2,:3,3 = 4(1− 2β ′1,1)(1− 2β ′2,2),:4,3 = 2(2β ′2,2 − 1),:3,4 = 2(2β ′1,1 − 1).

The component82 is separated into two parts8+2 and8−

2 according ast ≥ t0 ort ≤ t0. The imageT 8 in the target spaceR2, being a branch locus of the functionFβ ′(α1, α2; z1, z2), consists of 2 disjoint curvesT 81 andT 82 (see Figs. 2 and 3). Thecomponents of the complementR2

>0 −8 consist of three cells

;1 : ϕ(w) > 0, a1 > w1 > 0, a2 > w2 > 0,

;2 : ϕ(w) < 0, w1 > 0, w2 > 0,

;3 : ϕ(w) > 0, w1 > a1, w2 > a2.

T ;1 is the convex domain surrounded byT 81 including the origin.T ;3 is the domain surrounded byT 82. T ;2 is not planar and the domain overlap-

pingT ;3 surrounded byT 81, T 82, the negativez1 axis and the negativez2 axis.Consider the exterior domain surrounded byT 81 in the 1st octantz1 ≥ 0, z2 ≥ 0 in

the target space. We denote by;4 its inverse image byT in the complexificationC2 ofthe source space, such that81 is a boundary of;4.


0

0.02

0.04

0.06

0.08

0.1

0 0.02 0.04 0.06 0.08 0.1

"type 1(i)"

Fig. 1(a). β ′ :5 2

2 4

-1.25

-1.245

-1.24

-1.235

-1.23

-1.225

-1.22

-1.215

-1.21

-1.34 -1.335 -1.33 -1.325 -1.32 -1.315 -1.31 -1.305 -1.3

"t2-4a.p050"

Fig. 1(b)-1. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = 0.5


-1.31

-1.3

-1.29

-1.28

-1.27

-1.26

-1.34 -1.335 -1.33 -1.325 -1.32 -1.315 -1.31 -1.305 -1.3

"t2-4a.p015"

Fig. 1(b)-2. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = 0.15

-1.31

-1.3

-1.29

-1.28

-1.27

-1.26

-1.34 -1.335 -1.33 -1.325 -1.32 -1.315 -1.31 -1.305 -1.3

"t2-4a.p010"

Fig. 1(b)-3. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.1), s = 0.1


-1.32

-1.315

-1.31

-1.305

-1.3

-1.295

-1.29

-1.285

-1.28

-1.33 -1.325 -1.32 -1.315 -1.31 -1.305 -1.3 -1.295 -1.29

"t2-4a.p005"

Fig. 1(b)-4. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = 0.05

-1.33

-1.325

-1.32

-1.315

-1.31

-1.305

-1.3

-1.295

-1.29

-1.33 -1.325 -1.32 -1.315 -1.31 -1.305 -1.3 -1.295 -1.29

"t2-4a.p000"

Fig. 1(b)-5. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = 0.0


-1.33

-1.325

-1.32

-1.315

-1.31

-1.305

-1.3

-1.295

-1.29

-1.32 -1.315 -1.31 -1.305 -1.3 -1.295 -1.29 -1.285 -1.28

"t2-4a.m005"

Fig. 1(b)-6. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = −0.05

-1.33

-1.325

-1.32

-1.315

-1.31

-1.305

-1.3

-1.295

-1.29

-1.32 -1.315 -1.31 -1.305 -1.3 -1.295 -1.29 -1.285 -1.28

"t2-4a.m010"

Fig. 1(b)-7. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = −0.1


-1.34

-1.335

-1.33

-1.325

-1.32

-1.315

-1.31

-1.305

-1.3

-1.31 -1.305 -1.3 -1.295 -1.29 -1.285 -1.28 -1.275 -1.27

"t2-4a.m015"

Fig. 1(b)-8. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = −0.15

-1.35

-1.345

-1.34

-1.335

-1.33

-1.325

-1.32

-1.315

-1.31

-1.28 -1.275 -1.27 -1.265 -1.26 -1.255 -1.25 -1.245 -1.24

"t2-4a.m040"

Fig. 1(b)-9. β ′ :

1

20

1

2+ s

1

2

1

20

(−0.4≤ s ≤ 0.5), s = −0.4


-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03

"type 4(i)"

Fig. 1(c). β ′ : 50 50+ 5× 10−16

50+ 5× 10−16 50

-5

-4

-3

-2

-1

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4

"type 4(iv)"

Fig. 1(d). β ′ :0.5 1

1 0.5


Fig. 2. Fig. 3.

Lemma 10.;4 is homeomorphic to its image T ;4 which is itself a cell. It contains theboth real curves l+ × {w2 = 1}, and {w1 = 1} × l+ lying in the lines w2 = 1 andw1 = 1 respectively. ;4 is contained in the region ϕ(w1, w2) < 0 in a neighborhoodof the locus 81.

Proof. In fact, the JacobianJ of T is different from 0 inside of;4. Hence;4 ishomeomorphic to its imageT ;4. Let z = (η1t, η2t) t > 0 be a parametrization ofthe pointz in the target spaceR2, whereη1, η2 are non negative numbers such thatη1+ η2 = 1. We denote bytc the unique solution such that (η1t, η2t) lies inT 81. Thenwe have neart = tc the Puiseux expansions of the solution to (4.1),

w1 = ξ1 −√tc − tξ ′1 + · · · ,

w2 = ξ2 −√tc − tξ ′2 + · · · ,

where (ξ1, ξ2) is a point in81, and (ξ ′1, ξ ′2) (ξ ′1 > 0, ξ ′2 > 0) satisfies the linear equation

(ξ1 + β ′1,1(1− ξ1))ξ ′1ξ1+ β ′2,1(1− ξ1)

ξ ′2ξ2= 0,

β ′1,2(1− ξ2)ξ ′1ξ1+ (ξ2 + β ′2,2(1− ξ2))

ξ ′2ξ2= 0,

where its determinant vanishes. Hence we haveϕ(w1, w2) > 0 if t < tc and ϕ(w1, w2) < 0 if t > tc andt is neartc, sinceξ1+β ′1,1(1−ξ1) > 0, ξ2+β ′2,2(1−ξ2) > 0andξ1 > 1, ξ2 > 1.w lies in;4 if and only if t > tc. This proves Lemma 10."#

We now want to define several paths from the pointw = (1,1) in the complex affinespace ofw, along which the local solution (1) atz = (0,0) is continued analytically tothe other local solutions (2)–(9) expressed in Proposition 2 (see Fig. 4).

In the complex linew2 = 1 for z2 = 0, Eq. (4.1) reduces to

w1 − 1= z1wβ′1,1

1 . (5.5)


Fig. 4.

We take the real pathl+ in thew1 plane which is denoted byω2. Then the solution(1) goes to (2) owing to Lemma 5 in view ofβ ′1,1 > 1.

In the same way we can define the real pathω3 in the complex linew1 = 1, alongwhich (1) goes to (3). In a neighborhoodU2(δ1, δ2) of w = (0,1) for Z1 = 0, Eq. (4.1)reduces to

w2 − 1= Z2w

B

β′1,12 . (5.6)

Hence the real pathω2,4 = {w1 = 0} × l+ lying in the complex linew1 = 0 goes from(2) to (4) since B

β ′1,1> 1.

We also have (4) from (3) by interchanging the coordinatesw1, w2. We have thusdefined the pathω4 going from (1) to (4) as the compositeω1,2 ◦ ω2,4. ω5 denotes thepath fromw = (1,1) tow = (1,+∞) lying in the linew2 = 1 such thatw1 ∈ [1,∞)(see Lemma 4).

Then the local solution (1) goes to (5) alongω5. The pathω6 from (1) to (6) can bedefined similarly.

Let us get paths from (5) to (8) and from (5) to (9) respectively. Whenw1 tends to+∞ such thatZ2 is fixed,Z1 tends to 0, and thereforew2 in the solution (5) satisfies inU5(δ1, δ2)

w2 − 1= Z2w

β′2,2−B1−β′1,1

2 . (5.7)


Remark thatβ ′2,2−B1−β ′1,1 > 1. We take the pathω5,8 as{w1 = ∞}× l+ going from (5) to (8),

and the pathω5,9 as{w1 = +∞} × [1,+∞), where we have the solution (9) from (5).We can also define the pathsω6,7 andω6,9 getting (7) and (9) respectively from (6) by

interchangingw1 andw2. The composite of pathsω8 (or ω9)= ω5 ◦ω5,8 (or ω5 ◦ω5,9)goes from (1) to (8) (or (9)). We can define similarlyω′9 = ω6 ◦ ω6,9 going from (1) to(9).

Lemma 11. (1) Fβ ′ is continued analytically to Tσ1τ1Fβ ′ or Tσ2τ2Fβ ′ along T ω5 orT ω6.

(2) Fβ ′ is continued analytically to Tσ1σ2τ1τ2Fβ ′ along T ω9 or T ω′9.(3) Fβ ′ is continued analytically to Tτ1Fβ ′ ,Tτ2Fβ ′ , Tτ1σ2τ2Fβ ′ , Tτ2σ1τ1Fβ ′ along

T ω1, T ω2, T ω7, T ω8 respectively, where the paths should be taken in a detouredway around 8 in the complex affine space C2, such that argϕ(w) increases, whenthey cross 81 or 82.

(4) Fβ ′ is continued analytically to Tτ1τ2Fβ ′ along T ω4.

We denote byw(∞)0,0 the function (4) defined inU4(δ1, δ2).We denote further byS1 andS2 the shift operators obtained by analytic continuation

of rotation in the complex planez1 → e2πiz1 , z2 → e2πiz2 and put

w(∞)µ1,µ2= Sµ1

1 Sµ22 w

(∞)0,0 .

Crossing8 in the source space corresponds to a reflection onT 8 in the target space.Hence we have

Lemma 12.The reflection on T 81 in the target space gives rise to the transpositionbetween (1) and (5) or the one between (1) and (6). The reflection on T 8+

2 (or T 8−2 )

gives rise to the transposition between (5) and (9) (or the one between (6) and (9)).

We further define the pathT ω∗1 (orT ω∗2) as a path in the target space starting from theorigin, going throughT ;1, reflecting onT 8+

2 (or T 8−2 ) and going back to the origin.

The above lemma shows that the monodromy group generated by the pathsω∗1, ω∗2,ω5, ω6, gives rise to all the permutations among (1), (5), (6), (9), or equivalently amongFβ ′ , Tσ1τ1Fβ ′ , Tσ2τ2Fβ ′ , Tσ1σ2τ1τ2Fβ ′ . Hence, the monodromy group also gives rise to all

the permutations amongw(∞)0,0 , w(∞)−1,0, w

(∞)0,−1, w

(∞)−1,−1.

On the other hand,S1, S2 give the shifts for each component

S1 : w(∞)µ1,µ2→ w

(∞)µ1+1,µ2

, (5.8)

S2 : w(∞)µ1,µ2→ w

(∞)µ1,µ2+1. (5.9)

From now on we assume that

(C2) Let β = (β ′i,j ) be an arbitrary transform ofβ ′ by an element ofG. Thenboth

triples { β ′1,1, β ′2,1,1}, { β ′1,2, β ′2,2,1} are linearly independent over the field of rationals.

Letw(0)ν1,ν2 be the unique holomorphic solution at the origin to the following equations,corresponding to (1) in Proposition 2:

w1 − 1= z1e2πi(ν1β′1,1+ν2β

′2,1)w

β ′1,11 w

β ′2,12 ,

w2 − 1= z2e2πi(ν1β′1,2+ν2β

′2,2)w

β ′1,21 w

β ′2,22 . (5.10)


w = w(∞)µ1,µ2 has the similar expansion as (4.11)

w1 = e(−β′2,2+β′2,1

Bπ−2

β′2,2Bµ1π+2

β′2,1Bµ2π

)iZ1(1+ · · · ),

w2 = ei( β′1,2−β′1,1

Bπ+2

β′1,2Bµ1π−2

β′1,1Bµ2π

)iZ2(1+ · · · ).

To establish the relation of analytic continuation betweenw(0)ν1,ν2 andw(∞)µ1,µ2 forarbitrary (ν1, ν2) and (µ1, µ2) ∈ Z2, consider the real surfaceXC1,C2 (abbreviated byX) defined by (1.2).

Lemma 13.Suppose that

2πβ ′1,1ν1π + 2πβ ′2,1ν2π = ϕ1 + 2m1π, (5.11)

2πβ ′1,2ν1π + 2πβ ′2,2ν2π = ϕ2 + 2m2π, (5.12)

0< ϕ1 ≤ 2π , 0< ϕ2 ≤ 2π , (m1,m2)∈ Z2,

−β ′2,2 + β ′2,1B

π − 2β ′2,2Bµ1π + 2

β ′2,1Bµ2π = θ1 + 2l1π,

β ′1,2 − β ′1,1B

π + 2β ′1,2Bµ1π − 2

β ′1,1Bµ2π = θ2 + 2l2π, (5.13)

(l1, l2) ∈ Z2, −π < θ1 < π,−π < θ2 < π . We can put

C1 = 2πβ ′1,1ν1 + 2πβ ′2,1ν2, and C2 = 2πβ ′1,2ν1 + 2πβ ′2,2ν2.

w(0)ν1,ν2 is analytically continued tow(∞)µ1,µ2 along a path in the surfaceX if and only if

C1 − 2m1π = ϕ1 = π − β ′1,1θ1 − β ′2,1θ2,C2 − 2m2π = ϕ2 = π − β ′1,2θ1 − β ′2,2θ2, (5.14)

i.e., if and only if

µ1 + 1= −m1, µ2 + 1= −m2, ν1 = l1, ν2 = l2. (5.15)

Lemma 14.Let wν1,ν2 be the unique solution to the equations corresponding to (5) inProposition 2,

w1 − 1 = z1e2πiν2β′2,1w

β ′1,11 w

β ′2,12 ,

w2 − 1 = z2e2πiν2β′2,2w

β ′1,21 w

β ′2,22 , (5.16)

such that w1, w2 have the expansions at the origin

w1 = Z−11 e

2π(ν1+β′2,1ν2)i1−β′1,1 (1+ · · · ),

w2 = 1+ Z2e2πiν2β

′2,2+

2πi(ν1+β′2,1ν2)β′1,21−β′1,1 + · · · . (5.17)


We put

2π(ν1 + β ′2,1ν2)

1− β ′1,1= ϕ1 + 2m1π, 0 ≤ ϕ1 < 2π

and2π(ν1 + β ′2,1ν2)β

′1,2

1− β ′1,1+ 2πν2β

′2,2 = ϕ2 + 2m2π, 0< ϕ2 ≤ 2π.

wν,ν2 is analytically continued to w(∞)µ1,µ2 along a path in the surfaceX, if and only if

C1 − 2m1π = ϕ1 − β ′1,1(ϕ1 + 2m1π) = π − β ′1,1(θ1 + 2l1π)− β ′2,1θ2,C2 − 2m2π = ϕ2 − β ′1,2(ϕ1 + 2m1π) = π − β ′1,2(θ1 + 2l1π)− β ′2,2θ2, (5.18)

i.e.,

ν1 −m1 = µ1 + 1, −m2 = µ2 + 1, l2 = ν2. (5.19)

In the same way,

Lemma 15.Let wν1,ν2 be the unique solution to the equations corresponding to (6) inProposition 2,

w1 − 1 = z1e2πiν1β′1,1w

β ′1,11 w

β ′2,12 ,

w2 − 1 = z2e2πiν1β′1,2w

β ′1,21 w

β ′2,22 , (5.20)

such that w1, w2 have the expansions at the origin

w1 = 1+ Z1e2πiν1β

′1,1+

2πi(ν2+β′1,2ν1)β′2,11−β′2,2 + · · · ,

w2 = Z−12 e

2π(ν2+β′1,2ν1)i1−β′2,2 (1+ · · · ). (5.21)

Put2π(ν2 + β ′1,2ν1)β

′2,1

1− β ′2,2+ 2πν1β

′1,1 = ϕ1 + 2m1π, 0< ϕ1 ≤ 2π

and2π(ν2 + β ′1,2ν1)

1− β ′2,2= ϕ2 + 2m2π, 0 ≤ ϕ2 < 2π.

Letwν,ν2 be analytically continued tow(∞)µ1,µ2 along a path in the surfaceX. Then wehave

−m1 = µ1 + 1, ν2 −m2 = µ2 + 1, l1 = ν1. (5.22)


Lemma 16.Let w(0)ν1,ν2 = Sν11 S

ν22 w

(0)0,0 be the unique solution to Eq. (4.1)corresponding

to (9) in Proposition 2, such that we have the following expansions in U9(δ1, δ2)

w1 = e(2πν11−β′2,2G

+2πν2β′2,1G)iZ1(1+ · · · ),

w2 = e(2πν1β′1,2G+2πν2

1−β′1,1G

)iZ2(1+ · · · ). (5.23)

Put

2π(1− β ′2,2)ν1

G+ 2πβ ′2,1ν2

G= ϕ1 + 2m1π,

2π(β ′1,2)ν1

G+ 2π(1− β ′1,1)ν2

G= ϕ2 + 2m2π,

for 0 ≤ ϕ1 < 2π,0 ≤ ϕ2 < 2π, (m1, m2) ∈ Z2.Then w(0)ν,ν2 is analytically continued to w(∞)µ1,µ2 along a path in the surface X, if and

only if

C1 − 2m1π = ϕ1 − β ′1,1(ϕ1 + 2m1π)− β ′2,1(ϕ2 + 2m2π)

= π − β ′1,12π(θ1 + 2l1π)− β ′2,12π(θ2 + 2l2π),

C2 − 2m2π = ϕ2 − β ′1,2(ϕ1 + 2m1π)− β ′2,2(ϕ2 + 2m2π)

= π − β ′1,22π(θ1 + 2l1π)− β ′2,22π(θ2 + 2l2π), (5.24)

i.e.,

ν1 −m1 = µ1 + 1,

ν2 −m2 = µ2 + 1. (5.25)

Summing up these lemmas, we have shown that every local solution to (5.11), (5.17),(5.21) and (4.1) in the neighborhoodsU1(δ1, δ2), U5(δ1, δ2), U6(δ1, δ2), U9(δ1, δ2) inProposition 2 respectively can be analytically continued to one ofw

(∞)µ1,µ2 defined in the

neighborhoodU4(δ1, δ2). As a result we can conclude

Theorem 2.Under the condition (C1) and (C2) the monodromy group generated by T ω∗1,

T ω∗2, T ω5,T ω6, and S1, S2 contains every finite permutation among {w(∞)µ1,µ2}(−∞ <

µ1, µ2 <∞) and the shifts

S1 : w(∞)µ1,µ2→ w

(∞)µ1+1,µ2

,

S2 : w(∞)µ1,µ2→ w

(∞)µ1,µ2+1. (5.26)

The subgroup of finite permutations is normal in the group of all permutations so thatthe monodromy group is isomorphic to the semi-direct product of the subgroup of allfinite permutations and the 2 dimensional lattice Z2.

Remark 1. Our condition (ND) is not necessarily satisfied by an example arising inphysical problem (for example, Laughlin’s incompressible 1/m fluid explained in [15]or [26]).


6. Case Wheren is Arbitrary

The critical set8 of the map (1.1) is the complex algebraic hypersurface defined by

8 : ϕ(w1, . . . , wn) = 0. (6.1)

The set8 seems generally very complicated. To make it manageable, we assumenow that the matrixβ ′ has the following property:

(C3) The matrix β ′ − 1 is oscillatory i.e., all the subdeterminants of β ′ − 1 are non-negative and there exists a positive integer k such that the kth power (β ′ −1)k are totallypositive.

LetH = diag(η1, . . . , ηn) be the diagonal matrix such that allηj > 0. Then

Lemma 17.The equation

ϕ(η1t, . . . , ηnt) = 0 (6.2)

with respect to t has n positive roots which separate each other.

Proof. In fact, we have

ϕ(η1t, . . . , ηnt) = det(tH(1− β ′)+ β ′) = det(1− β ′) · det(tH − 1− (β ′ − 1)−1).

Sinceβ ′ − 1 is oscillatory, there exists a diagonal matrixJ with±1 entries such thatJ (β ′ − 1)−1J is oscillatory. HenceJ (1+ (β ′ − 1)−1)J is also. SinceH is a positivediagonal matrix,H−1J (1+ (β ′ −1)−1)J = JH−1(1+ (β ′ −1)−1)J is also oscillatory.This meansH−1(1+ (β ′ − 1)−1) has all positive eigenvalues which are different fromeach other.

We denote byt1, t2, . . . , tn then roots of (6.2) such that 0< t1 < · · · < tn and by8j the set of all points(η1tj , . . . , ηntj ), whereη1 ≥ 0, . . . , ηn ≥ 0 and

∑nj=1 ηj = 1.

Then

Corollary 2. In Rn≥0 the hypersurface 8 consists of n connected components 8j (1≤j ≤ n) which are non-singular.

We consider a pseudo-realn-cubeK with 2n vertices(v1, . . . , vn) in (CP 1)n suchthatvj is equal to 1 or+∞. The edges ofK are the paths of the formv1 × · · · vj−1 ×[1,+∞] × vj+1 × · · · × vn, where[1,+∞] denotes the interval from 0 to+∞ in thej th coordinateC plane.v1 × · · · vj−1 × [1,+∞] × vj+1 × · · · × vn meet8 at exactly one point. It meets

8j if and only if the number ofvk such thatvk = 1 is equal toj − 1.In the complex line defined byw1 = · · · = wp = +∞andwp+2 = · · · = wn = 1 the

corresponding solution to (1.1) is given by〈ε1, , . . . , εn〉 such thatε1 = · · · = εp = +∞andεp+2 = · · · = εn = 1, by puttingZ1 = · · · = Zp = Zp+2 = · · · = Zn = 0. From(2.1) and (2.2) we obtain the equation

wp+1 − 1= Zp+1wβ ′p+1,p+1p+1 , (6.3)

whereβ ′p+1,p+1 is defined as follows:


β ′p+1,p+1 = β ′p+1,p+1 − (β ′p+1,1, . . . , β′p+1,p) · (β ′p − 1)−1 · t (β ′1,p+1, . . . , (β

′p,p+1),

whereβ ′p denotes the submatrix(β ′i,j )pi,j=1.

We see thatβ ′p+1,p+1 > 1. This has the same form as (3.1), whence we can applythe casen = 1 to (6.3).

We call a path “admissible” if it has one of the following forms:

v1 × · · · vj−1 × l1 × vj+1 × · · · × vn,v1 × · · · vj−1 × l+ × vj+1 × · · · × vn,v1 × · · · vj−1 × l− × vj+1 × · · · × vn.

These paths will be abbreviated byej−1×l1×en−j ,ej−1×l+×en−j ,ej−1×l−×en−j(ek denotes thek products of the trivial pathe) respectively. They are all contained inX0,... ,0.

Lemma 18.〈1p,∞n−p〉 can be analytically continued to 〈1p−1,∞n−p+1〉 along thepath ep−1 × l1 × en−p.

For the proof see Lemma 4.In the same way

Lemma 19.〈0p,1q,∞n−p−q〉 is analytically continued to 〈0p+1,1q−1,∞n−p−q〉alongthe path ep × l+ × en−p−q .

For the proof see Lemma 5.

Lemma 20.〈0p,1q,∞n−p−q〉 is analytically continued to S−1p+q+1〈0p,1q,0,

∞n−p−q−1〉 along the path ep+q × l− × en−p−q−1.

For the proof see Lemma 6.The above three lemmas show the following:

Proposition 4. An arbitrary local solution 〈0p,1q,∞n−p−q〉 is an analytic continu-ation of 〈1n〉 along successive admissible paths in X0,... ,0, 〈0p,1q,∞n−p−q〉 can be

analytically continued to w(∞)ν1,... ,νn = S−1p+q+1 · · · S−1

n 〈0n〉 along successive admissiblepaths in X0,... ,0, where νj = 0 (1≤ j ≤ p + q) and νj = −1 (p + q + 1≤ j ≤ n).

On the other hand, in the complex linew1 = · · · = wp = 0, wp+1 = · · · =wp+q−1 = 1, wp+q+1 = · · · = wn = ∞, i.e., forZ1 = · · · = Zp+q−1 = Zp+q+1 =· · · = Zn = 0, Eq. (1.1) has the singularity at a pointZp+q = c(c > 1).

If we take, in theZp+q plane, the analytic continuation from 0 to 0 turning aroundccounterclockwise, then〈0p,1q,∞n−p−q〉and〈0p,1q−1,∞n−p−q+1〉are interchanged.Since〈0p,1q,∞n−p−q〉 goes to

w(∞)0×···×0︸︷︷︸p+q×

×−1×···×−1︸︷︷︸n−p−q×

by successive admissible paths, this fact is the same thing as interchanging

w(∞)0×···×0︸︷︷︸p+q×

×−1×···×−1︸︷︷︸n−p−q×


andw(∞)0×···×0︸︷︷︸p+q−1×

×−1×···×−1︸︷︷︸n−p−q+1×

.

Since this occurs also after any permutation of the coordinateswj , we have an arbi-

trary permutation amongw(∞)ν1,... ,νn for νj = 0, −1 by the above analytic continuation.Now we assume the following condition:

(C4) Let β ′ = (β ′i,j ) be an arbitrary transform of β ′ by an element of G. Then for

each k, the n + 1dimensional vector (1, β ′1,k, · · · , β ′n,k) are linearly independent overthe field of rationals.

Then we can conclude the following theorem in the same way as in the casen = 2.

Theorem 3.Under (C3) and (C4), the analytic continuation along the set of admissibleclosed paths in X0,... ,0 and the shifts Sj (1 ≤ j ≤ n) generate the monodromy groupfor the solution w(z) to (1.1). This group is isomorphic to the semi-direct product ofthe group of all finite permutations and the lattice group Zn among w(∞)ν1,... ,νn (−∞ <

ν1, . . . , νn <∞).Remark 2. The real varietyX = XC1,... ,Cn appearing in (1.2) plays an important partin studying the global nature of the solutions to Wu’s equations. Recently Kyoji Saitohas defined “real twisted forms” associated with real Coxeter arrangements and hasclarified their relationship to Coxeter groups (see [21]). In our investigation,X and themonodromy group in Theorem 2 seem to have a similar relation.

Acknowledgements. The authors appreciate the help of Prof. Takashi Sakajo to draw the figures, and usefuladvice from Prof. Takuo Fukuda about 2 dimensional mapping singularities. K. I. would like to thank TheMitsubishi Foundation for a scientific grant and Kazuko Iguchi for her continuous financial support andencouragement.

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