Table Of ContentANOTHER REFINEMENT OF THE BENDER-KNUTH (EX-)CONJECTURE
Ilse Fischer
Institut fu¨r Mathematik, Universit¨at Klagenfurt,
4 Universit¨atsstrasse 65-67, A-9020 Klagenfurt, Austria.
0
E-mail: [email protected]
0
2
n
a
J
9 Abstract. Wecomputethegeneratingfunctionofcolumn-strictplanepartitionswithpartsin
1
{1,2,...,n},atmostccolumns,prowsofoddlengthandk partsequalton. Thisrefinesboth,
] Krattenthaler’s[10] andthe author’s[5] refinementofthe Bender-Knuth(ex-)Conjecture. The
O result is proved by an extension of the method for proving polynomial enumeration formulas
C which was introduced by the author in [5] to q-quasi-polynomials.
.
h
t
a 1. Introduction
m
[ Let λ = (λ ,λ ,...,λ ) be a partition, i.e. λ ∈ Z and λ ≥ λ ≥ ... ≥ λ ≥ 0. A strict
1 2 r i 1 2 r
plane partition of shape λ is an array Π = (π ) of non-negative integers such that
1 i,j 1≤i≤r,1≤j≤λi
v the rows are weakly decreasing and the columns are strictly decreasing. For instance
5
3 7 6 5 5 2
2
5 4 2 2
1
0 4 2
4
2 1
0
/
h is a strict plane partition of shape (5,4,2,2). The norm n(Π) of a strict plane partition is
t
a defined as the sum of its parts and Π is said to be a strict plane partition of the non-negative
m
integer n(Π). Thus 47 is the norm of our example. Strict plane partitions and closely related
: objects have been enumerated subject to a variety of different constraints. In [2, p.50] Bender
v
i and Knuth had conjectured that the generating function with respect to the norm of strict
X
plane partitions with at most c columns and parts in {1,2,...,n} is equal to
r
a
n
[c+i;q]
qn(π) = i,
[i;q]
i
i=1
X Y
where[n;q] = 1+q+···+qn−1 = (1−qn)/(1−q)and[a;q] = n−1[a+i;q]. Thisconjecturewas
n i=0
proved by Andrews [1], Gordon [8], Macdonald [12, Ex. 19, p.53] and Proctor [13, Prop. 7.2].
Q
For related papers, which mostly include generalizations of the Bender-Knuth (ex-)Conjecture,
see [3, 4, 5, 9, 10, 14, 17].
In particular, Krattenthaler [10] computed the generating function of strict plane partitions
with parts in {1,2,...,n}, at most c columns and p rows of odd length. On the other hand the
author[5]computed thegeneratingfunctionofstrict planepartitionswithpartsin{1,2,...,n},
1
2 ILSEFISCHER
at most c columns and k parts equal to n. In this paper we refine these two results. The main
result is the following.
Theorem 1. The generating function of strict plane partitions with parts in {1,2,...,n}, at
most c columns, p rows of odd length and k parts equal to n is given by
M [k +1;q] [k −c−n+1;q] qk +L (−1)kqnk +(−1)nq(n−1)(2c+n)/2+k
n,c,p n−1 n−1 n,c,p
n−1
[k +1;q] [k −c−i+1;q] [k −c−n+1;q]
× (−1)cq(2i) n−1 i−1 n−i−1
[1;q] [1;q] [c+i+1;q]
i−1 n−1−i n−1
i=1
X
[k +1;q] [k +i+1;q] [k −c−n+1;q]
(i) i−1 n−i−1 n−1
−q 2
[1;q] [1;q] [c+i+1;q]
i−1 n−1−i n−1 !
where
(p+1) (p)
q 2 n−1 [c;q] q 2 n−1 [c+2n;q] n−1
p − p−1 [c+1;q]n−1[1;q]n−1 [c+2i+1;q]n−i 2|c
L = [c+hp;q]ni [hc+p+i1;q]n ! 2 i=1 [2i;q]n−i[2i;q]
n,c,p
q(p+21) n−1 −q(p2) n−1 [1;q]n−1 n−1 [c+2i;q]n−i Q 2 6 |c
p p−1 2 [2i;q]n−i[2i;q]
i=1
and (cid:16) h i h i(cid:17) Q
(−1)n−1q(n−1)(2c+n)/2
M =
n,c,p
[1;q]
n−2
(p+1) (p)
q 2 n−1 [c;q] q 2 n−1 n−1
p 1 − [c+2n−1;q] + p−1 [c+2n−1;q]2 [c+2i;q]n−i 2|c
× [c+hp;q]ni [n−1;q] [c+n;q][2n−2;q] [c+ph+1;q]ni[c+n;q][2n−2;q]!i=1 [2i;q]n−i .
q(p+21)+n−1 n−1(cid:16)+q(p2) n−1 1 (cid:17)n−1[c+2i+1;q]n−i−1 Q 2 6 |c
p p−1 [2n−2;q] [2i;q]n−i
i=1
(cid:16) h i h i(cid:17) Q
In these formulas the notion of the q-binomial coefficient is used. It is defined as follows.
n [n−k+1;q]k if 0 ≤ k ≤ n
= [1;q]k
k (0 otherwise
h i
At theend of Section 6 we show that Theorem 1 implies Krattenthaler’s and the author’srefine-
ment of the Bender-Knuth (ex-)Conjecture and with this the Bender-Knuth (ex-)Conjecture
itself.
Our method for proving Theorem 1 is an extension of the method for proving polynomial
enumeration formulas we have introduced in [5]. It is interesting to note that this elementary
method avoids the use of determinants completely, which is quite unusual in the field of plane
partition enumeration. The method is divided into the following three steps.
(1) Extension of the combinatorial interpretation. It only makes sense to ask for the
number of strict plane partitions with parts in {1,2,...,n}, at most c columns and k
parts equal to n if k ∈ {0,1,...,c}. This is because all n’s must be in the first row of the
strict plane partition by the columnstrictness. In the first step we find a combinatorial
extension of these strict plane partitions to arbitrary integers k, i.e. we find new objects
BENDER-KNUTH (EX-)CONJECTURE 3
indexed by an arbitrary integer k which are in bijection with strict plane partitions with
k parts equal to n if k ∈ {0,1,...,c}.
(2) The extending objects are enumerated by a q-quasi-polynomial in k. With
the help of a simple recursion we show that the extending objects are enumerated by a
q-quasi-polynomial (see Definition 1). Moreover the degree of the q-quasi-polynomial is
computed.
(3) Exploring properties of the q-quasi-polynomial that determine it uniquely.
A (q-quasi-)polynomial is determined by a finite number of properties such as zeros or
other evaluations. Inthe last stepwe derive enoughproperties oftheq-quasi-polynomial
in order to compute it using the degree estimation from the previous step.
Note that this article contains two types of extensions of the method for proving polynomial
enumeration formulas presented in [5]. Firstly, the method is extended to q-quasi-polynomials,
see Definition 1. More remarkable is, however, the following extension: In [5] we have described
a method that is applicable to polynomial enumeration formulas that factorize into distinct
linear factors over Z. There the “properties” in the third step are just the integer zeros together
with one (easy to compute) non-zero evaluation. (Thus the third step was entitled “Exploring
natural linear factors”.) In this article we demonstrate that the lack of enough integer zeros
can be compensated by other properties of the (q-quasi-)polynomial.
The paper is organized as follows. In Section 2 we give the combinatorial extension of
strict plane partitions as proposed in Step 1. In Section 3 we introduce the notion of q-quasi-
polynomials and establish the properties needed in this paper. In Section 4 we show that
the generating function of strict plane partitions which is under consideration in this paper
is a q-quasi-polynomial and we compute its degree (Step 2). In Section 5 we deduce enough
properties of the q-quasi-polynomial in order to compute it (Step 3). In Section 6 we perform
the (complicated) computation and in Section 7 we derive some q-summation formulas which
are needed in the computation.
Throughout the whole article we use the extended definition of the summation symbol,
namely,
f(a)+f(a+1)+···+f(b) if a ≤ b
b
f(i) = 0 if b = a−1 . (1.1)
Xi=a −f(b+1)−f(b+2)−···−f(a−1) if b+1 ≤ a−1
This assures that for any polynomial p(X) over an arbitrary integral domain I containing Q
there exists a unique polynomial q(X) over I such that y p(x) = q(y) for all integers y. We
x=0
usually write y p(x) for q(y).
x=0 P
P 2. Extension of the combinatorial interpretation
In this section we establish the combinatorial extension of strict plane partitions with parts
in{1,2,...,n}, at mostc columns andk partsequal to ntoarbitraryintegers k. This extension
was already introduced in [5, Section 2]. We repeat it here in less detail.
Let r,n,c be integers with 0 ≤ r ≤ n. A generalized (r,n,c) Gelfand-Tsetlin-pattern (for
short: (r,n,c)-pattern) is an array (a ) of integers with
i,j 1≤i≤r+1,i−1≤j≤n+1
(1) a = 0 and a = c,
i,i−1 i,n+1
(2) if a ≤ a then a ≤ a ≤ a
i,j i,j+1 i,j i−1,j i,j+1
4 ILSEFISCHER
(3) if a > a then a > a > a .
i,j i,j+1 i,j i−1,j i,j+1
The norm of an (r,n,c)-pattern is defined as the sum of its parts, where the first and the
last part of each row is omitted. A (3,6,c)-pattern for example is of the form
0 a a a c
4,4 4,5 4,6
0 a a a a c
3,3 3,4 3,5 3,6
0 a a a a a c
2,2 2,3 2,4 2,5 2,6
0 a a a a a a c,
1,1 1,2 1,3 1,4 1,5 1,6
such that every entry not in the top row is between its northwest neighbour w and its northeast
neighbour e, if w ≤ e then weakly between, otherwise strictly between. Thus
0 3 −5 10 4
0 2 −2 3 8 4
0 2 −1 2 4 7 4
0 0 0 1 2 5 6 4
is an example of an (3,6,4)-pattern. Note that a generalized (n − 1,n,c) Gelfand-Tsetlin-
pattern (a ) with 0 ≤ a ≤ c is what is said to be a Gelfand-Tsetlin-pattern with n rows
i,j n,n
and parts in {0,1,...,c}, see [16, p. 313] or [7, (3)] for the original reference. (Observe that
0 ≤ a ≤ c implies that the third condition in the definition of a generalized Gelfand-Tsetlin-
n,n
pattern never applies.) The following correspondence between Gelfand-Tsetlin-patterns and
strict plane partitions is crucial for our paper.
Lemma 1. There exists a norm-preserving bijection between Gelfand-Tsetlin-patterns with
n rows, parts in {0,1,...,c} and fixed a = k, and strict plane partitions with parts in
n,n
{1,2,...,n}, at most c columns and k parts equal to n. In this bijection (a ,a ,...,a )
1,n 1,n−1 1,1
is the shape of the strict plane partition.
Proof. Given such a Gelfand-Tsetlin-pattern, the corresponding strict plane partition is such
that the shape filled by parts greater than i corresponds to the partition given by the (n−i)-
th row (the top row being the first row) of the Gelfand-Tsetlin-pattern, where the first and
the last part of the row in the pattern are omitted. Thus the strict plane partition in the
introduction corresponds to the following Gelfand-Tsetlin pattern (first and last parts in the
rows are omitted).
1
0 2
0 1 4
0 1 2 4
0 0 1 2 4
0 0 1 2 4 5
0 0 0 1 2 4 5
(cid:3)
Therefore it suffices to compute the generating function with respect to the norm of (n −
1,n,c)-patterns with fixed a = k, 0 ≤ k ≤ c, and where exactly p values of a ,a ,...,a
n,n 1,1 1,2 1,n
BENDER-KNUTH (EX-)CONJECTURE 5
are odd. However, (n−1,n,c)-patterns are defined for all a ∈ Z and thus we have established
n,n
the combinatorial extension apart from the following technical detail. That is that we actually
have to work with a signed enumeration if a ∈/ {0,1,...,c}. Therefore we define the sign of
n,n
a pattern.
A pair (a ,a ) with a > a and i 6= 1 is called an inversion of the (r,n,c)-pattern
i,j i,j+1 i,j i,j+1
and (−1)#ofinversions is said to be the sign of the pattern, denoted by sgn(a). The (3,6,4)-
pattern in the example above has altogether 6 inversions and thus its sign is 1. We define the
following generating function
F (r,n,c,p;k ,k ,...,k ) = sgn(a)qnorm(a) /qk1+k2+...+kn−r,
q 1 2 n−r
!
a
X
where the sum is over all (r,n,c)-patterns (a ) with top row defined by k = a for
i,j i r+1,r+i
i = 1,...,n − r and such that exactly p of a ,a ,...,a are odd. It is crucial that for
1,1 1,2 1,n
0 ≤ k ≤ c F (n−1,n,c,p;k)qk is the generating function of (n−1,n,c)-patterns with a = k
q n,n
and where exactly p of a ,a ,...,a are odd. This is because an (n−1,n,c)-pattern with
1,1 1,2 1,n
0 ≤ a ≤ c has no inversions. Thus F (n−1,n,c,p;k) is the quantity we want to compute.
n,n q
It has the advantage that it is well defined for all integers k, whereas our original enumeration
problem was only defined for 0 ≤ k ≤ c.
3. q-quasi-polynomials and their properties
In the following let R be a ring containing C. A quasi-polynomial (see [15, page 210]) in the
variables X ,X ,...,X over R is an expression of the form
1 2 n
c (X ,X ,...,X )Xm1Xm2···Xmn,
m1,m2,...,mn 1 2 n 1 2 n
(m1,m2,...,Xmn)∈Zn,mi≥0
where (X ,X ,...,X ) → c (X ,X ,...,X ) are periodic functions on Zn taking
1 2 n m1,m2,...,mn 1 2 n
values in R, that is there exists an integer t with
c (k ,...,k ,...,k ) = c (k ,...,k +t,...,k )
m1,m2,...,mn 1 i n m1,m2,...,mn 1 i n
for all (k ,...,k ) ∈ Zn and i, and almost all c (X ,...,X ) are zero. Let (m ,...,m )
1 n m1,...,mn 1 n 1 n
be with c (X ,...,X ) 6= 0 such that m + ...+ m is maximal. Then m + ...+m
m1,...,mn 1 n 1 n 1 n
is said to be the degree of the quasi-polynomial. (The zero-quasi-polynomial is said to be of
degree −∞.) The smallest common period of all c (X ,...,X ) is said to be the period
m1,...,mn 1 n
of the quasi-polynomial. (In this paper we only deal with q-quasi-polynomials of period 1 or 2.)
In [5, Section 6] we have defined q-polynomials. The following definition of q-quasi-polynomials
is the merge of these two definitions. In this definition let R denote the ring of quotients with
q
elements from R[q] in the numerator and elements from C[q] in the denominator.
Definition 1. A q-quasi-polynomial over R in X ,X ,...,X is a quasi-polynomial over R
1 2 n q
in qX1,qX2,...,qXn. Let R [X ,X ,...,X ] denote the ring of these q-quasi-polynomials.
qq 1 2 n
Observe that R [X ,...,X ] is the ring of q-quasi-polynomials in X over
qq 1 n i
R [X ,...,X ,X ,...,X ].
qq 1 i−1 i+1 n
6 ILSEFISCHER
(Thus it would have been possible to define R [X ,...,X ] inductively with respect to n.) We
qq 1 n
define [X;q] = (1−qX)/(1−q) and [X;q] = n−1[X +i;q]. Observe that
n i=0
[X1;q]m1[X2;q]Qm2 ···[Xn;q]mn,
with (m ,m ,...,m ) ∈ Zn and m ≥ 0, is a basis of the q-quasi-polynomials over the periodic
1 2 n i
functions.
The following two properties of polynomials were crucial for our method for proving poly-
nomial enumeration formulas which we have introduced in [5]. Since we want to extend our
method to q-quasi-polynomials, we have to find q-quasi-analogs of these properties.
(1) If p(X) is a polynomial over R, then there exists a (unique) polynomial r(X) with
degr = degp+1 and
y
p(x) = r(y)
x=0
X
for every integer y.
(2) If p(X) is a polynomial over R and a is a zero of p(X), then there exists a polynomial
r(X) over R with
p(X) = (X −a)r(X).
Regarding the first property we show the following for q-quasi-polynomials.
Lemma 2. Let p(X) be a q-quasi-polynomial in X over R with degree d and period t. Then
y p(x)qx is a q-quasi-polynomial over R in y with degree at most d+1 and period at most
x=1
t.
P
In order to prove this lemma we need a definition and another lemma.
Definition 2. Let ρ → f(ρ) be a function. Then the q-differential-operator d is defined as
dqρ
follows
d f(qρ)−f(ρ)
f(ρ) = .
d ρ ρ(q −1)
q
With d we denote the n-fold application of the operator.
dqρn
Observe that for a laurent polynomial we have
c c
d
a ρi = [i;q]a ρi−1. (3.1)
i i
d ρ
q
i=b i=b
X X
Note that this is also true if b > c.
Lemma 3.
y
d σn−1((σq)y+1 −1)
[x;q] qxσx−1 =
n
d σn (σq −1)
x=0 q (cid:18) (cid:19)
X
Proof of Lemma 3. By (3.1) we have the following identity.
y y
d
[x;q] qxσx−1 = qxσx+n−1 .
n
d σn
q !
x=0 x=0
X X
BENDER-KNUTH (EX-)CONJECTURE 7
The assertion now follows from
y
σn−1((σq)y+1 −1)
qxσx+n−1 = .
(σq −1)
x=0
X
(cid:3)
Proof of Lemma 2. Suppose p(X) is a q-quasi-polynomial with period t. Let ρ ∈ C be a
primitive t-th root of unity. Then p(X) can be expressed as follows
p(X) = p (X)+ρXp (X)+ρ2Xp (X)+...+ρ(t−1)Xp (X),
0 1 2 t−1
where p (X) are q-polynomials, i.e. q-quasi-polynomials with period 1. Suppose d is the degree
i
of p(X). Then, for every i, we have
d
p (X) = a [X;q] ,
i i,j j
j=0
X
where a are coefficients in R . Thus, by Lemma 3,
i,j q
y y t−1 d t−1 d d σj−1((σq)y+1 −1)
p(x)qx = a [x;q] ρixqx = a ρi .
ij j i,j d σj (σq −1)
x=0 x=0 i=0 j=0 i=0 j=0 q (cid:18) (cid:19)(cid:12)σ=ρi
X XXX XX (cid:12)
(cid:12)
The assertion follows after observing that (cid:12)
d σj−1((σq)y+1 −1)
d σj (σq −1)
q (cid:18) (cid:19)(cid:12)σ=ρi
(cid:12)
is a q-quasi-polynomial in y of degree at most j +1. (cid:12) (cid:3)
(cid:12)
Next we consider the second important property of polynomials for our method. It suffices to
derive an analog for q-polynomials. Suppose p(X) is a q-polynomial over R and a is an integer
zero of p(X). Then there exists a q-polynomial r(X) over R with
p(X) = ([X;q]−[a;q])r(X) = qa[X −a;q]r(X).
The proof follows from the following identity
n−1 n−1
[X;q]n −[a;q]n = ([X;q]−[a;q]) [X;q]i[a;q]n−1−i = qa[X −a;q] [X;q]i[a;q]n−1−i.
i=0 i=0
X X
This property implies that for an integral domain R and distinct zeros a ,a ,...,a of the
1 2 r
q-polynomial p(X) there exists a q-polynomial r(X) with
r
p(X) = [X −a ;q] r(X).
i
!
i=1
Y
This will be fundamental for the “q-Lagrange interpolation” we use in Lemma 14.
8 ILSEFISCHER
4. F (n−1,n,c,p;k) is a q-quasi-polynomial in k
q
InthissectionweshowthatF (r,n,c,p;k ,...,k )isaq-quasi-polynomialink ,k ,...,k
q 1 n−r 1 2 n−r
with period 2. Moreover we show that the degree in k is at most 2r.
i
The following recursion is fundamental.
F (r,n,c,p;k ,k ,...,k ) =
q 1 2 n−r
k1 k2 k3 kn−r c
... F (r−1,n,c,p;l ,l ,...,l )ql1+l2+...+ln−r+1. (4.1)
q 1 2 n−r+1
lX1=0lX2=k1lX3=k2 ln−r=Xkn−r−1ln−r+X1=kn−r
Moreover we have
1 if exactly p of k ,k ,...,k are odd
1 2 n
F (0,n,c,p;k ,...,k ) =
q 1 n
0 otherwise
(
p n
e (k )
= 1,2 ij e (k ) =: S(n,p)(k ,...,k ),
0,2 j 1 n
e (k )
1≤i1<iX2<...<ip≤nYj=1 0,2 ij Yj=1
where x → e (x) is the function defined on integers with
i,t
1 x ≡ i mod t ρx −ρj
e (x) = = ,
i,t
0 otherwise ρi −ρj
(
0≤j≤t−1,j6=i
Y
where ρ ∈ C is a primitive t-th root of unity. The identity
F (0,n,c,p;k ,...,k ,...,k ) = F (0,n,c,p;k ,...,k +2,...,k )
q 1 i n q 1 i n
for all i, 1 ≤ i ≤ n, implies that F (0,n,c,p;k ,...,k ) is a q-quasi-polynomial with period 2.
q 1 n
The recursion (4.1) and Lemma 2 implies (inductively with respect to r) that F (r,n,c,p;.) is
q
a q-quasi-polynomial in (k ,k ,...,k ) with period at most 2.
1 2 n−r
For our purpose it is convenient to define the following generalization of F (r,n,c,p;.).
q
Definition 3. Let n,r, r ≤ n, be non-negative integers and A(k ,...,k ) a function on Zn.
1 n
We define G (r,n,c,A) inductively with respect to r: G (0,n,c,A) = A and
q q
G (r,n,c,A)(k ,...,k ) =
q 1 n−r
k1 k2 c
... G (r −1,n,c,A)(l ,l ,...,l )ql1+l2+...+ln−r+1 (4.2)
q 1 2 n−r+1
lX1=0lX2=k1 ln−r+X1=kn−r
With this definition we have
F (r,n,c,p;k ,...,k ) = G (r,n,c,S(n,p))(k ,...,k ).
q 1 n−r q 1 n−r
We define
T(n,i) = (−1)kj1+kj2+...+kji.
1≤j1<jX2<...<ji≤n
The following lemma shows that S(n,p) is a linear combination of T(n,1),T(n,2),...,T(n,n)
and T(n,0) := 1.
BENDER-KNUTH (EX-)CONJECTURE 9
Lemma 4.
n min(p,i)
1 i n−i
S(n,p) = (−1)l T(n,i)
2n l p−l
i=0 l=max(0,i−n+p) (cid:18) (cid:19)(cid:18) (cid:19)
X X
Proof. Set [n] := {1,2,...,n} and fix P ⊆ [n] with |P| = p. Then
1−(−1)kj 1+(−1)kj
e (k ) e (k ) = =
1,2 j 0,2 j
2 2
j∈P j∈[n]\P j∈P j∈[n]\P
Y Y Y Y
n min(p,i)
1
= (−1)l (−1)kj1+...+kjl (−1)km1+...+kmi−l,
2n
Xi=0 l=maxX(0,i−n+p) 1≤j1<X...<jl≤n, 1≤m1<X...<mi−l≤n,
jx∈P mx∈[n]\P
where the second equation follows by expanding the product. In the summation index i counts
the number of ±(−1)kx we choose from the product of the n factors of the form 1±(−1)kx and
the index l counts the number of −(−1)kx we choose. Observe that
i n−i
(−1)kj1+...+kjl (−1)km1+...+kmi−l = T(n,i),
l p−l
PX⊆[n], 1≤j1<X...<jl≤n, 1≤m1<X...<mi−l≤n, (cid:18) (cid:19)(cid:18) (cid:19)
|P|=p jx∈P mx∈[n]\P
because every (−1)kx1+...+kxi, 1 ≤ x1 < ... < xi ≤ n, appears with multiplicity il np−−li on the
left-hand-side, since there are i ways to choose the elements from {x ,...,x } =: I which lie
l 1 i (cid:0)(cid:1)(cid:0) (cid:1)
in P and n−i ways to choose the elements in [n]\I which lie in P. The assertion follows. (cid:3)
p−l (cid:0)(cid:1)
Lemma(cid:0)13(cid:1)from [5] implies that G (n − 1,n,c,1)(k) is a q-polynomial of degree 2n − 2 at
q
most in k. More general we aim to show that the degree of G (n − 1,n,c,T(n,p))(k) in k
q
is at most 2n − 2 as well. (Thus our result reproves Lemma 13 from [5].) The linearity of
A → G (r,n,c,A) and Lemma 4 then implies that the degree of G (n − 1,n,c,S(n,p)) is at
q q
most 2n−2 in k.
In fact we show that the degree of G (r,n,c,T(n,p)) in k is at most 2r. This degree
q i
estimation is rather complicated. Assume by induction with respect to r that the degree of
G (r −1,n,c,T(n,p))(k ,...,k ) in k is at most 2r −2 as well as the degree in k . The
q 1 n−r i i+1
degree of G (r,n,c,T(n,p)) in k is at most the degree of
q i
ki ki+1
G (r −1,n,c,T(n,p))(l ,...,l )
q 1 n−r+1
li=Xki−1li+X1=ki
in k with k = 0 and k = c. By Lemma 2 this allows us to conclude easily that the degree
i 0 n−r+1
of G (r,n,c,T(n,p)) in k is at most 4r −2, however, we want to establish that the degree is
q i
at most 2r. The following lemma is fundamental for this purpose. In order to state it we need
to define an operator D which is crucial for the analysis of (4.2).
i
Definition 4. Let G(k ,...,k ) be a function in m variables and 1 ≤ i ≤ m−1. We set
1 m
D G(k ,...,k ) =
i 1 m
G(k ,...,k ,k ,k ,k ,...,k )+G(k ,...,k ,k +1,k −1,k ,...,k ).
1 i−1 i i+1 i+2 m 1 i−1 i+1 i i+2 m
10 ILSEFISCHER
The following lemma shows the importance of this operator for the degree estimation.
Lemma 5. Let F(x ,x ) be a q-quasi-polynomial in x and x which is in x as well as in x
1 2 1 2 1 2
of degree at most R. Moreover assume that D F(x ,x ) is of degree at most R as a q-quasi-
1 1 2
polynomial in x and x , i.e. the linear combination of “monomials” [x ;q] [x ;q] ρx1ρx2 with
1 2 1 m 2 n 1 2
m + n ≤ R and where ρ and ρ are roots of unity. Then y b F(x ,x )qx1+x2 is of
1 2 x1=a x2=y 1 2
degree at most R+2 in y.
P P
Proof. Set F (x ,x ) = D F(x ,x )/2 and F (x ,x ) = (F(x ,x ) − F(x + 1,x − 1))/2.
1 1 2 1 1 2 2 1 2 1 2 2 1
Clearly F(x ,x ) = F (x ,x )+F (x ,x ). Observe that F (x +1,x −1) = −F (x ,x ). Thus
1 2 1 1 2 2 1 2 2 2 1 2 1 2
F (x ,x ) is a linear combination of expressions of the form
2 1 2
[x ;q] [x +1;q] ρx1−1ρx2 −[x ;q] [x +1;q] ρx2ρx1−1
1 m 2 n 1 2 1 n 2 m 1 2
with m,n ≤ R and where ρ and ρ are roots of unity . We set
1 2
d ρn−1((ρq)y+1 −1)
c(y,n,ρ) =
d ρn (ρq −1)
q (cid:18) (cid:19)
Lemma 3 implies
y b
([x ;q] [x +1;q] ρx1−1ρx2 −[x ;q] [x +1;q] ρx2ρx1−1)qx1+x2+1
1 m 2 n 1 2 1 n 2 m 1 2
xX1=axX2=y
= (c(y,m,ρ )−c(a−1,m,ρ ))(c(b+1,n,ρ )−c(y,n,ρ ))
1 1 2 2
−(c(y,n,ρ )−c(a−1,n,ρ ))(c(b+1,m,ρ )−c(y,m,ρ ))
2 2 1 1
= c(y,m,ρ )c(b+1,n,ρ )−c(a−1,m,ρ )c(b+1,n,ρ )+c(a−1,m,ρ )c(y,n,ρ )−
1 2 1 2 1 2
c(y,n,ρ )c(b+1,m,ρ )+c(a−1,n,ρ )c(b+1,m,ρ )−c(a−1,n,ρ )c(y,m,ρ ).
2 1 2 1 2 1
Observe that c(x,n,ρ ) is a q-quasi-polynomial in x of degree at most n+1 and thus
1
y b
F (x,y)qx+y
2
xX1=axX2=y
is of degree at most R+1 in y. By the assumption in the lemma y b F (x,y)qx+y is
of degree at most R+2 in y and the assertion follows. x1=a x2=y 1 (cid:3)
P P
Lemma 6. Let m be a positive integer, 1 ≤ i ≤ m and G(l) be a function in l = (l ,...,l ).
1 m
Then
k2 k3 km+1
D ... G(l ,...,l )
i 1 m
lX1=k1lX2=k2 lmX=km
1 k2 ki−1 ki+1+1 ki+1 ki+2 ki+3 km+1
= − ... ... D G(l)
i−1
2
lX1=k1 li−X2=ki−2li−1X=ki+1lXi=kili+1X=ki−1li+X2=ki+2 lmX=km
k2 ki−1 ki ki+1 ki+1−1 ki+3 km+1
+ ... ... D G(l) ,
i
lX1=k1 li−X2=ki−2li−X1=ki−1lXi=kili+1X=ki−1li+X2=ki+2 lmX=km
