Tài liệu Đề tài " Isomonodromy transformations of linear systems of difference equations" pptx

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1144 ALEXEI BORODIN
functions in the matrix elements of the initial conditions. This is our second
result.
In order to prove this claim, we introduce yet another set of coordinates
on A(z) with fixed A
0
, which is related to {B
i
} by a birational transformation.
It consists of matrices C
i
∈ Mat(m, C) with Sp(C
i
)=Sp(B
i
) such that
A(z)=A
0
(z −C
1
) ···(z −C
n
).
In these coordinates, the action of Z
n
is described by the relations
(7)

z +1−C
i

···

z +1−C
n

A
0

z −C
1

···

z −C
i−1

=

z +1−

C
i+1

···

z +1−

C
n

A
0

z −

C
1

···

z −

C
i

,
C
j
= C
j
(k
1
, ,k
n
),

C
j
= C
j
(k
1
, ,k
i−1
,k
i
+1,k
i+1
, ,k
n
) for all j.
Again, we prove that there exists a unique solution to these equations sat-
isfying Sp(C
i
(k)) = Sp(C
i
) − k
i
, for an arbitrary invertible A
0
and generic
{C
i
= C
i
(0)}. The solution is rational in the matrix elements of the initial
conditions.
The difference Schlesinger equations have an autonomous limit which con-
sists of (3), (4), and
(5-aut) B
i
(k
1
+1, ,k
n
+1)=A
−1
0
B
i
(k
1
, ,k
n
)A
0
,
(6-aut) Sp(B
i
(k
1
, ,k
n
)) = Sp(B
i
),i=1, ,n.
The equation (7) then becomes
(7-aut)

z −C
i

···

z −C
n

A
0

z −C
1

···

z −C
i−1

=

z −

C
i+1

···

z −

C
n

A
0

z −

C
1

···

z −

C
i

.
The solutions of these equations were essentially obtained in [V] via a gen-
eral construction of commuting flows associated with set-theoretical solutions
of the quantum Yang-Baxter equation; see [V] for details and references.
The autonomous equations can also be explicitly solved in terms of abelian
functions associated with the spectral curve {(z, w) : det(A(z) − wI)=0},
2
very much in the spirit of [MV, §1.5]. We hope to explain the details in a
separate publication.
The whole subject bears a strong similarity (and not just by name!) to the
theory of isomonodromy deformations of linear systems of differential equations
with rational coefficients:
dY(ζ)
dζ
=

B
∞
+
n

k=1
B
i
ζ − x
i

Y(ζ),(8)
2
It is easy to see that the curve is invariant under the flows.
ISOMONODROMY TRANSFORMATIONS
1145
which was developed by Schlesinger around 1912 and generalized by Jimbo,
Miwa, and Ueno in [JMU], [JM] to the case of higher order singularities. If
we analytically continue any fixed (say, normalized at a given point) solution
Y(ζ) of (8) along a closed path γ in C avoiding the singular points {x
k
} then
the columns of Y will change into their linear combinations: Y →YM
γ
. Here
M
γ
is a constant invertible matrix which depends only on the homotopy class
of γ. It is called the monodromy matrix corresponding to γ. The monodromy
matrices define a linear representation of the fundamental group of C with
n punctures. The basic isomonodromy problem is to change the differential
equation (8) so that the monodromy representation remains invariant.
There exist isomonodromy deformations of two types: continuous ones,
when x
i
move in the complex plane and B
i
= B
i
(x) form a solution of a sys-
tem of partial differential equations called Schlesinger equations, and discrete
ones (called Schlesinger transformations), which shift the eigenvalues of B
i
and
exponents of Y(ζ)atζ = ∞ by integers with the total sum of shifts equal to 0.
We prove that in the limit when
B
i
= x
i
ε
−1
+ B
i
,ε→ 0,
our action of Z
m(n+1)−1
in the discrete case converges to the action of Schlesinger
transformations on B
i
. This is our third result.
Furthermore, we argue that the “long-time” asymptotics of the Z
n
-action
in the discrete case (that is, the asymptotics of B
i
([x
1
ε
−1
], ,[x
n
ε
−1
])),
ε small, is described by the corresponding solution of the Schlesinger equa-
tions. More exactly, we conjecture that the following is true.
Take B
i
= B
i
(ε) ∈ Mat(m, C), i =1, ,n, such that
B
i
(ε) − y
i
ε
−1
+ B
i
→ 0,ε→ 0.
Let B
i
(k
1
, ,k
n
) be the solution of the difference Schlesinger equations (3.1)–
(3.3) with the initial conditions {B
i
(0) = B
i
}, and let B
i
(x
1
, ,x
n
)bethe
solution of the classical Schlesinger equations (5.4) with the initial conditions
{B
i
(y
1
, ,y
n
)=B
i
}. Then for any x
1
, ,x
n
∈ R and i =1, ,n,wehave
B
i

[x
1
ε
−1
], ,[x
n
ε
−1
]

+[x
i
ε
−1
]−y
i
ε
−1
+B
i
(y
1
−x
1
, ,y
n
−x
n
) → 0,ε→ 0.
In support of this conjecture, we explicitly show that the difference
Schlesinger equations converge to the conventional Schlesinger equations in
the limit ε → 0.
Note that the monodromy representation of π
1
(C \{x
1
, ,x
n
}) which
provides the integrals of motion for the Schlesinger flows, has no obvious analog
in the discrete situation. On the other hand, the obvious differential analog
of the periodic matrix P , which contains all integrals of motion in the case
of difference equations, gives only the monodromy information at infinity and
does not carry any information about local monodromies around the poles
x
1
, ,x
n
.
1146 ALEXEI BORODIN
Most of the results of the present paper can be carried over to the case
of q-difference equations of the form Y (qz)=A(z)Y (z). The q-difference
Schlesinger equations are, cf. (3)–(6),
(3q)
(4q)
(5q)
(6q)
B
i
( ) − B
i
( ,q
k
j
+1
, )=B
j
( ) − B
j
( ,q
k
i
+1
, ),
B
j
( ,q
k
i
+1
, )B
i
( )=B
i
( ,q
k
j
+1
, )B
j
( ),
B
i
(q
k
1
+1
, ,q
k
n
+1
)=q
−1
A
−1
0
B
i
(q
k
1
, ,q
k
n
)A
0
,
Sp(B
i
(q
k
1
, ,q
k
n
)) = q
−k
i
Sp(B
i
),i=1, ,n.
The q-analog of (7) takes the form
(7q)

z −q
−1
C
i

···

z −q
−1
C
n

A
0

z −C
1

···

z −C
i−1

=

z −q
−1

C
i+1

···

z −q
−1

C
n

A
0

z −

C
1

···

z −

C
i

,
C
j
= C
j
(q
k
1
, ,q
k
n
),

C
j
= C
j
(q
k
1
, ,q
k
i−1
,q
k
i
+1
,q
k
i+1
, ,q
k
n
) for all j.
A more detailed exposition of the q-difference case will appear elsewhere.
Similarly to the classical case, see [JM], discrete Painlev´e equations of
[JS], [Sak] can be obtained as reductions of the difference and q-difference
Schlesinger equations when both m (the size of matrices) and n (the degree
of the polynomial A(z)) are equal to two. For examples of such reductions
see [Bor, §3] for difference Painlev´e II equation (dPII), [Bor, §6] and [BB, §9]
for dPIV and dPV, and [BB, §10] for q-PVI. This subject still remains to be
thoroughly studied.
As was mentioned before, the difference and q-difference Schlesinger equa-
tions can be used to compute the gap probabilities for certain probabilistic
models. We conclude this introduction by giving an example of such a model.
We define the Hahn orthogonal polynomial ensemble as a probability measure
on all l-point subsets of {0, 1, ,N}, N>l>0, such that
Prob{(x
1
, ,x
l
)} = const ·

1≤i<j≤l
(x
i
− x
j
)
2
·
l

i=1
w(x
i
),
where w(x) is the weight function for the classical Hahn orthogonal polynomi-
als:
w(x)=

α + x
x

β + N − x
N − x

,α,β>−1orα, β < −N.
This ensemble came up recently in harmonic analysis on the infinite-dimensional
unitary group [BO, §11] and in a statistical description of tilings of a hexagon
by rhombi [Joh, §4].
The quantity of interest is the probability that the point configuration
(x
1
, ,x
l
) does not intersect a disjoint union of intervals [k
1
,k
2
]  ···
[k
2s−1
,k
2s
]. As a function in the endpoints k
1
, ,k
2s
∈{0, 1, ,N}; this
ISOMONODROMY TRANSFORMATIONS
1147
probability can be expressed through a solution of the difference Schlesinger
equations (3)–(6) for 2 × 2 matrices with n = deg A(z)=s +2,A
0
= I,
Sp(B
i
)={−k
i
, −k
i
},i=1, ,2s,
Sp(B
2s+1
)  Sp(B
2s+2
)={0, −α, N +1,N +1+β},
and with certain explicit initial conditions. The equations are also suitable for
numerical computations, and we refer to [BB, §12] for examples of those in the
case of a one interval gap.
I am very grateful to P. Deift, P. Deligne, B. Dubrovin, A. Its, D. Kazhdan,
I. Krichever, G. Olshanski, V. Retakh, and A. Veselov for interesting and
helpful discussions.
This research was partially conducted during the period the author served
as a Clay Mathematics Institute Long-Term Prize Fellow.
1. Birkhoff ’s theory
Consider a matrix linear difference equation of the first order
Y (z +1)=A(z)Y (z).(1.1)
Here A : C → Mat(m, C) is a rational function (i.e., all matrix elements of
A(z) are rational functions of z) and m ≥ 1. We are interested in matrix
meromorphic solutions Y : C → Mat(m, C) of this equation.
Let n be the order of the pole of A(z) at infinity, that is,
A(z)=A
0
z
n
+ A
1
z
n−1
+ lower order terms .
We assume that (1.1) has a formal solution of the form
Y (z)=z
nz
e
−nz

ˆ
Y
0
+
ˆ
Y
1
z
+
ˆ
Y
2
z
2
+

diag

ρ
z
1
z
d
1
, ,ρ
z
m
z
d
m

(1.2)
with ρ
1
, ,ρ
m
= 0 and det
ˆ
Y
0
=0.
3
It is easy to see that if such a formal solution exists then ρ
1
, ,ρ
m
must
be the eigenvalues of A
0
, and the columns of
ˆ
Y
0
must be the corresponding
eigenvectors of A
0
.
Note that for any invertible T ∈ Mat(m, C), (TY)(z) solves the equation
(TY)(z +1)=(TA(z)T
−1
)(TY)(z).
Thus, if A
0
is diagonalizable, we may assume that it is diagonal without loss
of generality. Similarly, if A
0
= I and A
1
is diagonalizable, we may assume
that A
1
is diagonal.
3
Substituting (1.2) in (1.1) we use the expansion

z+1
z

nz
= e
nz ln(1+z
−1
)
= e
n
−
ne
n
2z
+
to compare the two sides.
1148 ALEXEI BORODIN
Proposition 1.1. If A
0
= diag(ρ
1
, ,ρ
m
), where {ρ
i
}
m
i=1
are nonzero
and pairwise distinct, then there exists a unique formal solution of (1.1) of the
form (1.2) with
ˆ
Y
0
= I.
Proof. It suffices to consider the case n = 0; the general case is reduced
to it by considering (Γ(z))
n
Y (z) instead of Y (z), because
Γ(z)=
√
2πz
z−
1
2
e
−z

1+
1
12
z
−1
+

.
(More precisely, this expression formally solves Γ(z +1)=zΓ(z).)
Thus, we assume n = 0. Then we substitute (1.2) into (1.1) and compute
ˆ
Y
k
one by one by equating the coefficients of z
−l
, l =0, 1, .If
ˆ
Y
0
= I then
the constant coefficients of both sides are trivially equal. The coefficients of
z
−1
give
ˆ
Y
1
A
0
+ diag(ρ
1
d
1
, ,ρ
m
d
m
)=A
0
ˆ
Y
1
+ A
1
.(1.3)
This equality uniquely determines {d
i
} and the off-diagonal entries of
ˆ
Y
1
,be-
cause
[
ˆ
Y
1
,A
0
]
ij
=(ρ
j
− ρ
i
)(
ˆ
Y
1
)
ij
.
Comparing the coefficients of z
−2
we obtain
(
ˆ
Y
2
−
ˆ
Y
1
)A
0
+
ˆ
Y
1
diag(ρ
1
d
1
, ,ρ
m
d
m
)+ = A
0
ˆ
Y
2
+ A
1
ˆ
Y
1
+ ,
where the dots stand for the terms which we already know (that is, those
which depend only on ρ
i
’s, d
i
’s, A
i
’s, and
ˆ
Y
0
= I). Since the diagonal values of
A
1
are exactly ρ
1
d
1
, ρ
n
d
n
by (1.3), we see that we can uniquely determine
the diagonal elements of
ˆ
Y
1
and the off-diagonal elements of
ˆ
Y
2
from the last
equality.
Now let us assume that we already determined
ˆ
Y
1
, ,
ˆ
Y
l−2
and the off-
diagonal entries of
ˆ
Y
l−1
by satisfying (1.1) up to order l − 1. Then comparing
the coefficients of z
−l
we obtain
(
ˆ
Y
l
−(l −1)
ˆ
Y
l−1
)A
0
+
ˆ
Y
l−1
diag(ρ
1
d
1
, ,ρ
m
d
m
)+ = A
0
ˆ
Y
l
+ A
1
ˆ
Y
l−1
+ ,
where the dots denote the terms depending only on ρ
i
’s, d
i
’s, A
i
’s, and
ˆ
Y
0
, ,
ˆ
Y
l−2
. This equality allows us to compute the diagonal entries of Y
l−1
and the off-diagonal entries of Y
l
. Induction on l completes the proof.
The condition that the eigenvalues of A
0
are distinct is not necessary for
the existence of the asymptotic solution, as our next proposition shows.
Proposition 1.2. Assume that A
0
= I and A
1
= diag(r
1
, ,r
n
) where
r
i
−r
j
/∈{±1, ±2, } for all i, j =1, ,n. Then there exists a unique formal
solution of (1.1) of the form (1.2) with
ˆ
Y
0
= I.
ISOMONODROMY TRANSFORMATIONS
1149
Proof. As in the proof of Proposition 1.1, we may assume that n =0.
Comparing constant coefficients we see that ρ
1
= ···= ρ
m
= 1. Then equating
the coefficients of z
−1
we find that d
i
= r
i
, i =1, ,m. Furthermore, equating
the coefficients of z
−l
, l ≥ 2 we find that
[
ˆ
Y
l−1
,A
1
] − (l −1)
ˆ
Y
l−1
is expressible in terms of A
i
’s and
ˆ
Y
1
, ,
ˆ
Y
l−2
. This allows us to compute all
ˆ
Y
i
’s recursively.
We call two complex numbers z
1
and z
2
congruent if z
1
− z
2
∈ Z.
Theorem 1.3 (G. D. Birkhoff [Bi1, Th. III]). Assume that
A
0
= diag(ρ
1
, ,ρ
m
),
ρ
i
=0,i=1, ,m, ρ
i
/ρ
j
/∈ R for all i = j.
Then there exist unique solutions Y
l
(z)(Y
r
(z)) of (1.1) such that:
(a) The function Y
l
(z)(Y
r
(z)) is analytic throughout the complex plane ex-
cept possibly for poles to the right (left ) of and congruent to the poles of
A(z)(respectively, A
−1
(z −1));
(b) In any left (right ) half-plane Y
l
(z)(Y
r
(z)) is asymptotically represented
by the right-hand side of (1.2).
Remark 1.4. Part (b) of the theorem means that for any k =0, 1, ,





Y
l,r
(z) z
−nz
e
nz
diag(ρ
−z
1
z
−d
1
, ,ρ
−z
m
z
−d
m
) −
ˆ
Y
0
−
ˆ
Y
1
z
−···−
ˆ
Y
k−1
z
k−1





≤
const
z
k
for large |z| in the corresponding domain.
Theorem 1.3 holds for any (fixed) choices of branches of ln(z) in the left
and right half-planes for evaluating z
−nz
= e
−nz ln(z)
and z
−d
k
= e
−d
k
ln(z)
, and
of a branch of ln(ρ) with a cut not passing through ρ
1
, ,ρ
m
for evaluating
ρ
−z
k
= e
−z ln ρ
k
. Changing these branches yields the multiplication of Y
l,r
(z)
by a diagonal periodic matrix on the right.
Remark 1.5. Birkhoff states Theorem 1.3 under a more general assump-
tion: he only assumes that the equation (1.1) has a formal solution of the form
(1.2). However, as pointed out by P. Deligne, Birkhoff’s proof has a flaw in
case one of the ratios ρ
i
/ρ
j
is real. The following counterexample was kindly
communicated to me by Professor Deligne.
Consider the equation (1.1) with m = 2 and
A(z)=

11/z
01/e

.
1150 ALEXEI BORODIN
The formal solution (1.2) has the form
Y (z)=

I +

0 a
00

z
−1
+

10
0 e
−z

with a = e/(1 − e).
Actual solutions that we care about have the form
Y (z)=

1 u(z)
0 e
−z

where u(z) is a solution of u(z +1)=u(z)+e
−z
/z. In a right half-plane we
can take
u
r
(z)=−
∞

n=0
e
−(z+n)
z + n
.
The first order approximation of u
r
(z) anywhere except near nonpositive inte-
gers is
u
r
(z) ∼−
∞

n=0
e
−(z+n)
z
=
ae
−z
z
.
Next, terms can be obtained by expanding 1/(z + n).
In order to obtain a solution which behaves well on the left, it suffices to
cancel the poles:
u
l
(z)=u
r
(z)+
2πi
e
2πiz
− 1
.
The corresponding solution Y
l
(z) has the needed asymptotics in sectors of the
form π/2+ε<arg z<3π/2+ε, but it has the wrong asymptotic behavior as
z → +i∞. Indeed, lim
z→+i∞
u
l
(z)=−2πi.
On the other hand, we can take
u
l
(z)=u
l
(z)+2πi = u
r
(z)+
2πi e
2πiz
e
2πiz
− 1
,
which has the correct asymptotic behavior in π/2 − ε<arg z<3π/2 −ε, but
fails to have the needed asymptotics at −i∞.
Remark 1.6. In the case when |ρ
1
| > |ρ
2
| > ··· > |ρ
m
| > 0, a result
similar to Theorem 1.3 was independently proved by R. D. Carmichael [C].
He considered the asymptotics of solutions along lines parallel to the real axis
only. Birkhoff also referred to [N] and [G] where similar results had been proved
somewhat earlier.
Now let us restrict ourselves to the case when A(z) is a polynomial in z.
The general case of rational A(z) is reduced to the polynomial case by the
following transformation. If (z − x
1
) ···(z − x
s
) is the common denominator
of {A
kl
(z)} (the matrix elements of A(z)), then
¯
Y (z)=Γ(z −x
1
) ···Γ(z −x
s
) · Y (z)
ISOMONODROMY TRANSFORMATIONS
1151
solves
¯
Y (z +1)=
¯
A(z)
¯
Y (z) with polynomial
¯
A(z)=(z − x
1
) ···(z −x
s
)A(z).
Note that the ratio P (z)=(Y
r
(z))
−1
Y
l
(z) is a periodic function. (The
relation P (z +1) = P (z) immediately follows from the fact that Y
l,r
solves
(1.1).) From now on let us fix the branches of ln(z) in the left and right half-
planes mentioned in Remark 1.4 so that they coincide in the upper half-plane.
Then the structure of P (z) can be described more precisely.
Theorem 1.7 ([Bi1, Th. IV]). With the assumptions of Theorem 1.3,
the matrix elements p
kl
(z) of the periodic matrix P(z)=(Y
r
(z))
−1
Y
l
(z) have
the form
p
kk
(z)=1+c
(1)
kk
e
2πiz
+ ···+ c
(n−1)
kk
e
2π(n−1)iz
+ e
2πid
k
e
2πniz
,
p
kl
(z)=e
2πλ
kl
z

c
(0)
kl
+ c
(1)
kl
e
2πiz
+ ···+ c
(n−1)
kl
e
2π(n−1)iz

(k = l),
where c
(s)
kl
are some constants, and λ
kl
denotes the least integer as great as the
real part of (ln(ρ
l
) − ln(ρ
k
))/2πi.
Thus, starting with a matrix polynomial A(z)=A
0
z
n
+A
1
z
n−1
+···+A
n
with nondegenerate A
0
= diag(ρ
1
, ,ρ
m
), ρ
k
= ρ
l
for k = l, we construct the
characteristic constants {d
k
}, {c
(s)
kl
} using Proposition 1.1 and Theorems 1.3,
1.7.
Note that the total number of characteristic constants is exactly the same
as the number of matrix elements in matrices A
1
, ,A
n
. Thus, it is natural
to ask whether the map
(A
1
, ,A
n
) →

{d
k
}, {c
(s)
kl
}

is injective or surjective (the constants ρ
1
, ,ρ
n
being fixed). The following
partial results are available.
Theorem 1.8 ([Bi2, §17]). For any nonzero ρ
1
, ,ρ
m
, ρ
i
/ρ
j
/∈ R for
i = j, there exist matrices A
1
, ,A
n
such that the equation (1.1) with A
0
=
diag(ρ
1
, ρ
m
) either possesses the prescribed characteristic constants {d
k
},
{c
(s)
kl
}, or else constants {d
k
+ l
k
}, {c
(s)
kl
}, where l
1
, ,l
m
are integers.
Theorem 1.9 ([Bi1, Th. VII]). Assume there are two matrix polynomi-
als A

(z)=A

0
z
n
+ ···+ A

n
and A

(z)=A

0
z
n
+ ···+ A

n
with
A

0
= A

0
= diag(ρ
1
, ρ
m
),ρ
k
=0,ρ
k
/ρ
l
/∈ R for k = l,
such that the sets of the characteristic constants for the equations Y

(z +1)=
A

(z)Y

(z) and Y

(z +1) = A

(z)Y

(z) are equal. Then there exists a rational
matrix R(z) such that
A

(z)=R(z +1)A

(z)R
−1
(z),
1152 ALEXEI BORODIN
and the left and right canonical solutions Y
l,r
of the second equation can be
obtained from those of the first equation by multiplication by R on the left :
(Y

)
l,r
= R (Y

)
l,r
.
2. Isomonodromy transformations
The goal of this section is to construct explicitly, for given A(z), ratio-
nal matrices R(z) such that the transformation A(z) → R(z +1)A(z)R
−1
(z),
cf. Theorem 1.9 above, preserves the characteristic constants (more generally,
preserves {c
(s)
kl
} and shifts d
k
’s by integers).
Let A(z) be a matrix polynomial of degree n ≥ 1, A
0
= diag(ρ
1
, ,ρ
m
),
and ρ
i
’s are nonzero and their ratios are not real. Fix mn complex numbers
a
1
, ,a
mn
such that a
i
− a
j
/∈ Z for any i = j. Denote by M(a
1
, ,a
mn
;
d
1
, ,d
m
) the algebraic variety of all n-tuples of m by m matrices A
1
, ,A
n
such that the scalar polynomial
det A(z) = det(A
0
z
n
+ A
1
z
n−1
+ ···+ A
n
)
of degree mn has roots a
1
, ,a
mn
, and ρ
i

d
i
−
n
2

=(A
1
)
ii
(this comes from
the analog of (1.3) for arbitrary n).
Theorem 2.1. For any κ
1
, ,κ
mn
∈ Z, δ
1
, ,δ
m
∈ Z,
mn

i=1
κ
i
+
m

j=1
δ
j
=0,
there exists a nonempty Zariski open subset A of M(a
1
, ,a
mn
; d
1
, ,d
m
)
such that for any (A
1
, ,A
n
) ∈Athere exists a unique rational matrix R(z)
with the following properties:

A(z)=R(z +1)A(z)R
−1
(z)=

A
0
z
n
+

A
1
z
n−1
+ ···+

A
n
,

A
0
= A
0
,
(

A
1
, ,

A
n
) ∈M(a
1
+ κ
1
, ,a
mn
+ κ
mn
; d
1
+ δ
1
, ,d
m
+ δ
m
),
and the left and right canonical solutions of

Y (z +1) =

A(z)

Y (z) have the
form

Y
l,r
= RY
l,r
,
where Y
l,r
are left and right canonical solutions of Y (z +1)=A(z)Y (z).
The map (A
1
, ,A
n
) → (

A
1
, ,

A
n
) is a birational map of algebraic
varieties.
Remark 2.2. The theorem implies that the characteristic constants {c
(s)
kl
}
for the difference equations with coefficients A and

A are the same, while the
constants d
k
are being shifted by δ
k
∈ Z.
ISOMONODROMY TRANSFORMATIONS
1153
Note also that if we require that all d
k
’s do not change, then, by virtue of
Theorem 1.9, Theorem 2.1 provides all possible transformations which preserve
the characteristic constants. Indeed, if A

(z)=R(z+1)A

(z)R
−1
(z) then zeros
of det A

(z) must be equal to those of det A

(z) shifted by integers.
Proof. Let us prove the uniqueness of R first. Assume that there exist two
rational matrices R
1
and R
2
with needed properties. This means, in particular,
that the determinants of the matrices

A
(1)
= R
1
(z +1)A(z)R
−1
1
(z) and

A
(2)
= R
2
(z +1)A(z)R
−1
2
(z)
vanish at the same set of mn points a
i
= a
i
+κ
i
, none of which are different by
an integer. Denote by

Y
r
1
= R
1
Y
r
and

Y
r
2
= R
2
Y
r
the right canonical solutions
of the corresponding equations. Then

Y
r
1
(

Y
r
2
)
−1
= R
1
R
−1
2
is a rational matrix
which tends to I at infinity. Moreover,


Y
r
1
(

Y
r
2
)
−1

(z +1)=

A
(1)
(z)


Y
r
1
(

Y
r
2
)
−1

(z)


A
(2)
(z)

−1
.
Since

Y
r
1
(

Y
r
2
)
−1
is holomorphic for z  0, the equation above implies that
this function may only have poles at the points which are congruent to a
i
(zeros
of det

A
(2)
(z)) and to the right of them. (Recall that two complex numbers
are congruent if their difference is an integer.) But since

Y
r
1
(

Y
r
2
)
−1
is also
holomorphic for z  0, the same equation rewritten as


Y
r
1
(

Y
r
2
)
−1

(z)=


A
(1)
(z)

−1


Y
r
1
(

Y
r
2
)
−1

(z +1)

A
(2)
(z)
implies that this function may only have poles at the points a
i
(zeros of
det

A
(1)
(z)) or at the points congruent to them and to the left of them. Thus,

Y
r
1
(

Y
r
2
)
−1
= R
1
R
−1
2
is entire, and by Liouville’s theorem it is identically equal
to I. The proof of uniqueness is complete.
To prove the existence we note, first of all, that it suffices to provide a
proof if one of the κ
i
’s is equal to ±1 and one of the δ
j
’s is equal to ∓1 with
all other κ’s and δ’s equal to zero. The proof will consist of several steps.
Lemma 2.3. Let A(z) be an m by m matrix-valued function holomorphic
near z = a, and det A(z)=c(z − a)+O

(z −a)
2

as z → a, where c =0.
Then there exists a unique (up to a constant) nonzero vector v ∈ C
m
such that
A(a)v =0. Furthermore, if B(z) is another matrix-valued function which is
holomorphic near z = a, then (BA
−1
)(z) is holomorphic near z = a if and
only if B(a)v =0.
Proof. Let us denote by E
1
the matrix unit which has 1 as its (1, 1)-entry
and 0 as all other entries. Since det A(a) = 0, there exists a nondegenerate
constant matrix C such that A(a)CE
1
= 0 (the first column of C must be a
0-eigenvector of A(a)). This implies that
H(z)=A(z)C(E
1
(z −a)
−1
+ I −E
1
)