Table Of ContentA reduced fast construction of polynomial lattice
point sets with low weighted star discrepancy
Ralph Kritzinger∗, Helene Laimer†, Mario Neumüller‡
7
1
Abstract
0
2
The weighted star discrepancy is a quantitative measure for the performance of
n
point sets in quasi-Monte Carlo algorithms for numerical integration. We con-
a
J sider polynomial lattice point sets, whose generating vectors can be obtained by
0 a component-by-component construction to ensure a small weighted star discre-
1 pancy. Our aim is to significantly reduce the construction cost of such generating
vectors by restricting the size of the set of polynomials from which we select the
]
T
components of the vectors. To gain this reduction we exploit the fact that the
N
weights of the spaces we consider decay very fast.
.
h
Keywords: weighted star discrepancy, polynomial lattice point sets,
t
a
quasi-Monte Carlo integration, component-by-component algorithm
m
MSC 2000: 11K06, 11K38, 65D30, 65D32
[
1
1 Introduction
v
5
2
5 A convenient way to approximate the value of an integral
2
0
I (F) := F(x)dx
. s
1 Z[0,1)s
0
7 over the s-dimensional unit cube is to use a quasi-Monte Carlo rule of the form
1
:
v 1 N−1
Xi QN,s(F) := N F(xn). (1)
n=0
X
r
a
The integrand F usually stems from some suitable (weighted) function space and the
multiset P of integration nodes x ,x ,...,x in the algorithm Q (F) is chosen
0 1 N−1 N,s
deterministically from [0,1)s. For comprehensive information on quasi-Monte Carlo
algorithms consult, e.g., [5, 3, 9, 12]. The quality of a quasi-Monte Carlo rule is for in-
stance measured by some notion of discrepancy. In this paper we consider the weighted
star discrepancy, which has been introduced by Sloan and Woźniakowski in [21], ex-
ploiting the insight that the weights reflect the influence of different coordinates on the
∗R. Kritzinger is supported by the Austrian Science Fund (FWF): Project F5509-N26, which is a
part of the Special Research Program"Quasi-Monte Carlo Methods: Theory and Applications".
†H. Laimer is supported by the Austrian Science Fund (FWF): Project F5506-N26,which is a part
of the Special Research Program"Quasi-Monte Carlo Methods: Theory and Applications".
‡M. Neumüller is supported by the Austrian Science Fund (FWF): Project F5505-N26, which is a
part of the Special Research Program"Quasi-Monte Carlo Methods: Theory and Applications".
1
integration error. Let [s] := {1,2,...,s} and consider a weight sequence γ = (γu)u⊆[s]
of nonnegative real numbers, i.e., every group of variables (xi)i∈u is equipped with a
weight γu. Roughly speaking, a small weight indicates that the corresponding variables
contribute little to the integration problem. For simplicity, throughout this paper we
only consider product weights, defined as follows. Given a non-increasing sequence of
positive real numbers (γj)j≥1 with γj ≤ 1 we set γu := j∈uγj and γ∅ := 1.
Definition 1 Let γ = (γu)u⊆[s] be a weight sequence anQd P = {x0,...,xN−1} ⊆ [0,1)s
be an N-element point set. The local discrepancy of the point set P at t = (t ,...,t ) ∈
1 s
(0,1]s is defined as
1 N−1 s
∆(t,P) := 1 (x )− t ,
[0,t) n j
N
n=0 j=1
X Y
where 1[0,t) denotes the characteristic function of [0,t) := [0,t1) × ··· × [0,ts). The
weighted star discrepancy of P is then defined as
DN∗,γ(P) := t∈s(u0,p1]s∅m6=ua⊆x[s]γu|∆((tu,1),P)|,
where (tu,1) denotes the vector (t˜1,...,t˜s) with t˜j = tj if j ∈ u and t˜j = 1 if j ∈/ u.
A relation between the integration error of quasi-Monte Carlo rules and the weighted
star discrepancy is given by the Koksma-Hlawka type inequality (see [21])
|Q (F)−I (F)| ≤ D∗ (P)kFk ,
N,s s N,γ γ
where k·k is some norm which depends only on the weight sequence γ but not on the
γ
point set P.
It turns out that lattice point sets (see, e.g., [12, Chapter 5], [9]) and polynomial
lattice point sets (see, e.g., [12, Chapter 4], [13], [5, Chapter 10]) are often a good choice
as sample points in (1). These two kind of point sets are strongly connected and have a
lot of parallel tracks in their analysis. However, there are some situations were one type
of point set is superior to the other in terms of error bounds or the size of the function
classes where they yield good results for numerical integration. Thus it is beneficial to
have constructions for lattice point sets as well as for polynomial lattice point sets at
hand. For a detailed comparison of lattice point sets and polynomial lattice point sets
see, e.g., [19]. Also, Ch. Schwab, inresponsetothefirst author’stalkaboutconstructing
lattice point sets at the MCQMC 2016 conference in Stanford, pointed out that it would
be an interesting problem to extend the result in [7] to polynomial lattice point sets.
Thus, in this paper we study polynomial lattice point sets, a special class of point sets
with low weighted star discrepancy, introduced by Niederreiter in [12, Chapter 4], [13].
For a prime number p, let F be the finite field of order p. We identify F with the set
p p
{0,1,...,p − 1} equipped with the modulo p arithmetic. We denote by F [x] the set
p
of polynomials over F and by F ((x−1)) the field of formal Laurent series over F with
p p p
elements of the form
∞
L = t x−l,
l
l=ω
X
where ω ∈ Z and t ∈ F for all l ≥ ω. For a given dimension s ≥ 2 and some integer
l p
m ≥ 1 we choose a so-called modulus f ∈ F [x] with deg(f) = m as well as polynomials
p
2
g ,...,g ∈ F [x]. The vector g = (g ,...,g ) is called the generating vector of the
1 s p 1 s
polynomial lattice point set. Further, we introduce the map φ : F ((x−1)) → [0,1)
m p
such that
∞ m
φ t x−l = t p−l.
m l l
!
lX=ω l=mXax{1,ω}
With n ∈ {0,1,...,pm −1} we associate the polynomial
m−1
n(x) = n xr ∈ F [x],
r p
r=0
X
as each such n can uniquely be written as n = n + n p+ ···+n pm−1 with digits
0 1 m−1
n ∈ {0,1,...,p−1} for all r ∈ {0,1,...,m−1}. With this notation, the polynomial
r
lattice point set P(g,f) is defined as the set of N := pm points
n(x)g (x) n(x)g (x)
x = φ 1 ,...,φ s ∈ [0,1)s
n m m
f(x) ! f(x) !!
for 0 ≤ n ≤ pm −1. See also [5, Chapter 10].
Inthefollowing, byG we denote theset ofallpolynomials g over F with deg(g) <
p,m p
m. Further we define
G (f) := {g ∈ G | gcd(g,f) = 1}. (2)
p,m p,m
For the weighted star discrepancy of a polynomial lattice point set we simply write
D∗ (g,f).
N,γ
Niederreiter [12] proved the existence of polynomial lattice point sets with low un-
weighted star discrepancy by averaging arguments. Generating vectors of good polyno-
mial lattice point sets can be constructed by a component-by-component (CBC) con-
struction. The standard structure of CBC constructions is as follows. We start by
setting the first coordinate of the generating vector equal to 1. After this first step we
proceed by increasing the dimension of the generating vector by one in each step until
we have a generating vector (g ,...,g ) of full size s. That is, all previously chosen
1 s
components stay the same and one new component is added. This new coordinate is
chosen from a given search set, most commonly from G (f) given by (2). Usually it
p,m
is determined such that the weighted star discrepancy of the lattice point set, corres-
ponding to the generating vector, consisting of all previously chosen components plus
one additional component, is minimized as a function of this last component.
Such constructions were provided in [4] for an irreducible modulus f and in [1] for
a reducible f. In these papers, the authors considered the unweighted star discrepancy
as well as its weighted version, which we study here. It is the aim of the present paper
to speed up these constructions by reducing the search sets for the components of the
generating vector g according to each component’s importance. It is the nature of
product weighted spaces that the components g of the generating vector have less and
j
less influence on the quality of the corresponding polynomial lattice point as j increases.
Roughly speaking this is due to the weights (γ ) that are becoming ever smaller with
j
increasing index j. We want to exploit this property in the following way. As the
components’ influence is decreasing with their indices we want to use less and less time
and computational cost to choose these components. To achieve this we choose them
3
from smaller and smaller search sets, which are defined as follows. Let w ≤ w ≤ ··· be
1 2
a non-decreasing sequence of nonnegative integers. This sequence of w ’s is determined
j
in accordance with the weight sequence γ. Loosely speaking, the smaller γ , the bigger
j
w is chosen. For w ∈ N with w < m we define G and G (f) analogously to
j 0 p,m−w p,m−w
G and G (f), respectively. Further we set
p,m p,m
G (f) ifw < m,
p,m−w
G (f) :=
p,m−w {1 ∈ F [x]} ifw ≥ m
p
for any w ∈ N . For w < m these sets have cardinality pm−w − 1 in the case of an
0
irreducible modulus f and pm−w−1(p−1) for the special case f : F → F ,x 7→ xm. We
p p
will consider these two cases in what follows. Finally, for d ∈ [s], we define Gd (f) :=
p,m−w
G (f)×···×G (f). The idea is to choose the ith component of g of the form
p,m−w1 p,m−wd
xwig , where g ∈ G (f), i.e., the search set for the ith component is reduced by a
i i p,m−wi
factor p−min{wi,m} in comparison to the standard CBC construction. We will show that
under certain conditions on the weights γ and the parameters w a polynomial lattice
i
point set constructed according to our reduced CBC construction has a low weighted
star discrepancy of order N−1+δ for all δ > 0. The standard CBC construction (cf.
[20]) can be done in O(sN2) operations. To speed up the construction, in a first step,
making use of ideas from Nuyens and Cools [17, 18] on fast Fourier transformation, the
construction cost can be reduced to O(sN logN), as for example done in [4]. Combining
this with our reduced search sets we obtain a computational cost that is independent
of the dimension eventually. Reduced CBC constructions have been introduced first
by Dick et al. in [2] for lattice and polynomial lattice point sets with a small worst
case integration error in Korobov and Walsh spaces, respectively, and have also been
investigated in [7] for lattice point sets with small weighted star discrepancy.
An interesting aspect of the discrepancy of high dimensional point sets is the so-
called tractability of discrepancy (see, e.g., [14, 15, 16] for detailed information). For
N,s ∈ N let
disc (N,s) := inf D∗ (P),
∞ N,γ
P⊆[0,1)s
#P=N
the Nth minimal star discrepancy. To introduce the concept of tractability of discrep-
ancy we define the information complexity (also called the inverse of the weighted star
discrepancy) as
N∗(s,ε) := min{N ∈ N | disc (N,s) ≤ ε}.
∞
Thus N∗(s,ε) is the minimal number of points required to achieve a weighted star
discrepancy of at most ε. To keep the construction cost of our generating vector low, it
is, ofcourse, beneficialtohaveasmallinformationcomplexityandthustostandachance
to have a polynomial lattice point set of small size. This is why we are interested in
how fast the information complexity grows when s and ε−1 tend to infinity. Tractability
describes this dependence of the information complexity on the dimension s and the
error demand ε. The best we can hope for is the case where N∗(s,ε) is independent of s
and depends at most polynomially on ε−1. To be more precise, we say that we achieve
strong polynomial tractability if there exist constants C,τ > 0 such that
N∗(s,ε) ≤ Cε−τ
4
for all s ∈ N and all ε ∈ (0,1). Roughly speaking, a problem is considered tractable if its
information complexity’s dependence on s and ε−1 is not exponential. Taking weights
into account in the definition of discrepancy can sometimes overcome the so-called curse
of dimensionality, i.e., an exponential dependence of N∗(s,ε) on s. We will show that
our reduced fast CBC algorithm finds a generating vector g of a polynomial lattice point
set that achieves strong polynomial tractability provided that
∞
γ pwj < ∞
j
j=1
X
with a construction cost of
min{s,t}
O N +min{s,t}N +N (m−w )p−wd
d
d=1
X
operations, where t = max{j ∈ N | w < m}.
j
Before stating our main results we would like to discuss a motivating example. Consider
first the standard CBC construction as treated in [1, 4], where w = 0 for all j ≥ 0. In
j
this case, a sufficient condition for strong polynomial tractability is ∞ γ < ∞, which
j=1 j
for instance is satisfied for the special choices γ = j−2 and γ = j−1000. However, in the
j j P
second example the weights decay much faster than in the first. We can make use of this
fact by introducing the sequence w = (w ) such that the condition ∞ γ pwj < ∞
j j≥0 j=1 j
holds, while still achieving strong polynomial tractability (see Corollary 2). This way,
P
we can reduce the size of the search sets for the components of the generating vector if
the weights γ decay very fast. Consider for example the weight sequence γ = j−k for
j j
some k > 1. For w = ⌊(k −α)log j⌋ with arbitrary 1 < α < k we find
j p
∞ ∞ ∞
γ pwj ≤ j−kjk−α = j−α = ζ(α) < ∞,
j
j=1 j=1 j=1
X X X
where ζ denotes the Riemann Zeta function. Observe that for large k, i.e., fast decaying
weights, we may choose smaller search sets and thereby speed up the CBC algorithm.
This paper is organized as follows. In the next section we give an algorithm for con-
structing polynomial lattice point sets and we derive an upper bound on the weighted
star discrepancy of the point set constructed with this algorithm. We also give tractab-
ility results and analyze the computational cost of our algorithm. At first, we consider
the case where f : F → F ,x 7→ xm. Then we consider the case where the modulus of
p p
the polynomial lattice point set is irreducible.
2 A reduced CBC construction
In this section we present a CBC construction for the vector (xw1g ,...,xwsg ) and an
1 s
upper bound for the weighted star discrepancy of the corresponding polynomial lattice
point set.
First note that if g ∈ Gs , then it is known (see [4]) that
p,m
1 |u|
DN∗,γ(g,f) ≤ uX⊆[s]γu 1−(cid:18)1− N(cid:19) !+Rγs(g,f), (3)
u6=∅
5
where in the case of product weights we have
s
Rs(g,f) = r (h ,γ ). (4)
γ p i i
h∈GXsp,m\{0}iY=1
h·g≡0modf
Here, for elements h = (h ,...,h ) and g = (g ,...,g ) in Gs we define the scalar
1 s 1 s p,m
product by h·g := h g +···+h g . The numbers r (h,γ) for h ∈ G and γ ∈ R are
1 1 s s p p,m
defined as
1+γ if h = 0,
r (h,γ) =
p γr (h) otherwise,
p
where for h = h +h x+···+h xa with h 6= 0 we set
0 1 a a
1
r (h) = .
p
pa+1sin2 πh
p a
(cid:16) (cid:17)
Thus, in order to analyze the weighted star discrepancy of a polynomial lattice point
set it suffices to investigate the quantity Rs(g,f). This is due to the result of Joe [10],
γ
who proved that for any summable weight sequence (γ ) we have
j j≥1
1 |u| max(1,Γ)e ∞i=1γi
γu 1− 1− ≤ ,
uX⊆[s] (cid:18) N(cid:19) ! N P
u6=∅
with Γ := ∞ γi .
i=1 1+γi
P
Algorithm 1 Let p ∈ P, m ∈ N, f ∈ F [x] and let (w ) be a non-decreasing sequence
p j j≥1
of nonnegative integers and consider product weights (γ ) . Construct (g ,...,g ) ∈
j j≥1 1 s
Gs (f) as follows:
p,m−w
1. Set g = 1.
1
2. For d ∈ [s − 1] assume (g ,...,g ) ∈ Gd (f) to be already found. Choose
1 d p,m−w
g ∈ G (f) such that
d+1 p,m−wd+1
Rd+1(xw1g ,...,xwdg ,xwd+1g )
γ 1 d d+1
is minimized as a function of g .
d+1
3. Increase d by 1 and repeat the second step until (g ,...,g ) is found.
1 s
Remark 1 Of course we have Gs (f) ⊆ Gs , and thus in Algorithm 1 it indeed
p,m−w p,m
suffices to consider Rd+1 rather than the weighted star discrepancy.
γ
In the algorithm above, the search set is reduced for each coordinate of (g ,...,g )
1 s
according to its importance, as with increasing w the search set becomes smaller, as the
j
weight γ and thus the corresponding component’s influence on the quality of the gen-
j
erating vector decreases. For this reason we call Algorithm 1 a reduced CBC algorithm.
We will now study Algorithm 1 for different choices of f.
6
m
2.1 Polynomial lattice point sets for f(x) = x
We will now study the interesting case where f: F → F ,x 7→ xm. Throughout the
p p
rest of this section we write xm instead of f to emphasize our special choice of f. Note
that for g ∈ F ((x−1)) the Laurent series g/f can be easily computed in this case by
p
shifting the coefficients of g m times to the left. It is the aim of this section to prove
the following theorem:
Theorem 1 Let γ = (γ ) and w with 0 = w ≤ w ≤ ···. Let further (g ,...,g ) ∈
j j≥1 1 2 1 s
Gs (xm) be constructed using Algorithm 1. Then we have for every d ∈ [s]
p,m−w
1 d p2 −1
Rd((xw1g ,...,xwdg ),xm) ≤ 1+γ +γ 2pmin{wi,m}m .
γ 1 d pm i i 3p !
i=1
Y
As a direct consequence we obtain the following discrepancy estimate.
Corollary 1 Let N = pm and γ, w and (g ,...,g ) as in Theorem 1. Then the poly-
1 s
nomial lattice point set P((xw1g ,...,xwsg ),xm) has a weighted star discrepancy
1 s
D∗ ((xw1g ,...,xwsg ),xm)
N,γ 1 s
1 |u| 1 s p2 −1
≤ γu 1− 1− + 1+γi +γi2pmin{wi,m}m . (5)
uX⊆[s] (cid:18) N(cid:19) ! N iY=1 3p !
u6=∅
Knowing the above discrepancy bound, we are now ready to ask about the size of
the polynomial lattice point set required to achieve a weighted star discrepancy not
exceeding some ε threshold. In particular, we would like to know how this size depends
on the dimension s and on ε.
Corollary 2 Let N = pm, γ and w as in Theorem 1 and consider the problem of
constructing generating vectors for polynomial lattice point sets with small weighted star
discrepancy. Then
∞
γ pwj < ∞
j
j=1
X
is a sufficient condition for strong polynomial tractability. This condition further implies
D∗ ((xw1g ,...,xwsg ),xm) = O(N−1+δ), with the implied constant independent of s,
N,γ 1 s
for any δ > 0, where (g ,...,g ) ∈ Gs (xm) is constructed using Algorithm 1.
1 s p,m−w
Proof. Construct a generating vector (g ,...,g ) ∈ Gs (xm) applying Algorithm 1
1 s p,m−w
and consider its weighted star discrepancy, bounded by (5). Following closely the lines
of the argumentation in [7, Section 5] and noticing that 2mp2−1 = O(logN) we obtain
3p
the result. More precisely, provided that the γ pwj’s are summable, we have a means to
j
construct polynomial lattice point sets P(g,f) with D∗ (g,f) ≤ ε, whose sizes grow
N,γ
polynomially in ε−1 and are independent of the dimension. As a result the problem is
strongly polynomially tractable. The discrepancy result D∗ ((xw1g ,...,xwsg ),xm) =
N,γ 1 s
O(N−1+δ) also follows directly from [7]. ✷
7
Remark 2 Recall that t = max{j ∈ N: w < m} and note that setting w = m
j j
for all j > t does neither change the bound on the weighted star discrepancy nor the
computational cost of Algorithm 1. It might change the generating vector though. If so,
however, only components with very little influence on the quality of the point set are
altered. Defining w = m for all j > t, it suffices to have a summable weight sequence
j
γ in order to achieve strong polynomial tractability, as long as t is finite.
In order to show Theorem 1 we need several auxiliary results.
Lemma 1 Let a ∈ F [x] be monic. Then we have
p
p2 −1
r (h) = (m−deg(a)) p−deg(a).
p
3p
h∈GXp,m\{0}
a|h
In particular, for a = 1 this formula yields
p2 −1
r (h) = m .
p
3p
h∈GXp,m\{0}
Proof. This fact follows from [1, p. 1055] (by setting γ = 1). The special case a = 1
d+1
also follows from [4, Lemma 2.2] by setting s = 1. ✷
For our purposes, it is convenient to write Rs(g,f) from (4) in an alternative way.
γ
To this end, we introduce some notation. For a Laurent series L ∈ F ((x−1)) we denote
p
by c (L) its coefficient of x−1, i.e., its residuum. Further, we set X (L) := χ (c (L)),
−1 p p −1
where χ is a non-trivial additive character ofF . One could for instance choose χ (n) =
p p p
e2pπin for n ∈ F (see, e.g., [11]). It is clear that X (L) = 1 if L is a polynomial and that
p p
X (L +L ) = X (L )X (L ) for L ,L ∈ F ((x−1)). From [12, p. 78] we know that
p 1 2 p 1 p 2 1 2 p
v pm if f | g,
X g = (6)
p f ! 0 otherwise.
v∈XGp,m
Lemma 2 We have
s 1 s v
Rs(g,f) = − (1+γ )+ 1+γ +γ r (h)X hg .
γ i pm i i p p f i!
iY=1 v∈XGp,miY=1 h∈GXp,m\{0}
Proof. We employ the properties of X as stated above to obtain from (4)
p
s 1 s v
Rs(g,f) =− (1+γ )+ r (h ,γ ) X h·g
γ i pm p i i ! p f !
iY=1 h∈XGsp,m iY=1 v∈XGp,m
s 1 s v
=− (1+γ )+ r (h ,γ )X h g
i pm p i i p f i i!
iY=1 v∈XGp,miY=1 hi∈XGp,m
s 1 s v
=− (1+γ )+ 1+γ +γ r (h)X hg ,
i pm i i p p f i!
iY=1 v∈XGp,miY=1 h∈GXp,m\{0}
✷
and the claimed formula is verified.
8
Now we study a sum which will appear later in the proof of Theorem 1 and show an
upper bound for it.
Lemma 3 Let w ∈ N and v ∈ G . Let
0 p,m
v
Y (v,xm) := r (h)X hxwg ,
pm,w p p xm
g∈Gp,mX−w(xm)h∈GXp,m\{0} (cid:18) (cid:19)
where xw denotes the polynomial f(x) = xw. Then we have
1 p2 −1
|Y (v,xm)| ≤ 2pmin{w,m}m .
#G (xm) pm,w 3p
p,m−w v∈XGp,m
Proof. Let us first assume that w ≥ m. Then we have G (xm) = {1} and therefore
p,m−w
p2 −1
Y (v,xm) = r (h)X (vhxw−m) = r (h) = m
pm,w p p p
3p
h∈GXp,m\{0} h∈GXp,m\{0}
with Lemma 1. This leads to
1 p2 −1 p2 −1
|Y (v,xm)| = pmm ≤ 2pmin{w,m}m
#G (xm) pm,w 3p 3p
p,m−w v∈XGp,m
in this case. For the rest of the proof let w < m and additionally we abbreviate
#G (xm) by #G. We write
p,m−w
1 1 1
|Y (v,xm)| = |Y (v,xm)|+ |Y (v,xm)|.
pm,w pm,w pm,w
#G #G #G
v∈XGp,m v∈XGp,m v∈XGp,m
xm−w|v xm−w∤v
In what follows, we refer to the latter sums as
1 1
S := |Y (v,xm)| and S := |Y (v,xm)|.
1 pm,w 2 pm,w
#G #G
v∈XGp,m v∈XGp,m
xm−w|v xm−w∤v
We may uniquely write any v ∈ G \{0} in the form v = qxm−w+ℓ, where q,ℓ ∈ F [x]
p,m q
with deg(q) < w and deg(ℓ) < m − w. Using the properties of X it is clear that
p
Y (v,xm) = Y (ℓ,xm) and hence
pm,w pm,w
1 1
S = |Y (0,xm)| = r (h)
1 pm,w p
#G #G
v∈XGp,m v∈XGp,m g∈Gp,mX−w(xm)h∈GXp,m\{0}
xm−w|v xm−w|v
p2 −1 p2 −1
= m = pmin{w,m}m .
3p 3p
v∈XGp,m
xm−w|v
We move on to S . Let e(ℓ) := max{k ∈ {0,1,...,m−w −1} : xk | ℓ}. With this
2
definition we may display S as
2
pw m−w−1
S = |Y (ℓ,xm)|. (7)
2 pm,w
#G
kX=0 ℓ∈Gp,Xm−w\{0}
e(ℓ)=k
9
In the following, we compute Y (ℓ,xm) for ℓ ∈ G \{0} with e(ℓ) = k. Let µ be
pm,w p,m−w p
theMöbius functionontheset ofmonicpolynomialsover F , i.e., µ : F [x] → {−1,0,1}
p p p
and
(−1)ν if h is squarefree and has ν irreducible factors,
µ (h) =
p 0 else.
The fact that µp(1)= 1, µp(x) = −1 and µp(xi) = 0 for i ∈ N, i ≥ 2, yields the
equivalence of µ (t) = 1 and gcd(xm−w,g) = 1. Therefore we can write
t|gcd(xm−w,g) p
P
ℓ
Y (ℓ,xm) = r (h) X hg µ (t)
pm,w p p xm−w ! p
h∈GXp,m\{0} g∈GXp,m−w t|gcd(Xxm−w,g)
ℓ
= r (h) µ (t) X hg
p p p xm−w !
h∈GXp,m\{0} t|xXm−w g∈GXp,m−w
t|g
ℓ
= r (h) µ (t) X hat
p p p xm−w !
h∈GXp,m\{0} t|xXm−w a∈Gp,mX−w−deg(t)
xm−w a
= r (h) µ X hℓ
p p p
h∈GXp,m\{0} t|xXm−w t !a∈GXp,deg(t) (cid:18)t (cid:19)
xm−w
= r (h) µ pdeg(t)
p p
t !
h∈GXp,m\{0} t|xXm−w
t|hℓ
xm−w
= µ pdeg(t) r (h).
p p
t !
t|xXm−w h∈GXp,m\{0}
t|hℓ
The equivalence of the conditions t | hℓ and t | h yields
gcd(t,ℓ)
xm−w
Y (ℓ,xm) = µ pdeg(t) r (h).
pm,w p p
t !
t|xXm−w h∈GXp,m\{0}
t |h
gcd(t,ℓ)
We investigate the inner sum and use Lemma 1 with a = t to find
gcd(t,ℓ)
t p2 −1
r (h) = m−deg p−deg(gcd(tt,ℓ)).
p
gcd(t,ℓ)!! 3p
h∈GXp,m\{0}
t |h
gcd(t,ℓ)
Now we have
p2 −1 xm−w t
Y (ℓ,xm) = µ m−deg pdeg(gcd(t,ℓ))
pm,w p
3p t ! gcd(t,ℓ)!!
t|xXm−w
p2 −1 xm−w
= m µ pdeg(gcd(t,ℓ))
p
3p t !
t|xXm−w
p2 −1 xm−w t
− µ deg pdeg(gcd(t,ℓ)).
p
3p t ! gcd(t,ℓ)!
t|xXm−w
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