Table Of ContentRATIONALITY OF ZETA FUNCTIONS OVER FINITE FIELDS
SUNWOOPARK
Abstract. The zeta function of an affine variety over a finite field contains
comprehensive information on the number of points of the variety for all the
field extensions of the finite field. This expository paper follows Koblitz’s
treatment[2]ofDwork’sproof[1]oftherationalityofzetafunctionsofaffine
varietiesoverfinitefields,alsoknownastherationalitystatementoftheWeil
Conjectures.
Contents
1. Introduction 1
2. Power Series over C 3
p
2.1. Logarithms, Exponents, and Other Power Series 4
2.2. Newton Polygons 7
2.3. Overconvergent Power Series 11
3. Rationality of Zeta Functions 14
3.1. Zeta Functions over Finite Fields 14
3.2. p-adic Meromorphic 16
3.3. Rationality 20
Acknowledgments 24
References 24
1. Introduction
Let f(x ,...,x ) be a polynomial with n variables over a finite field F where
1 n q
q =pr is a power of a prime p. The natural question then arises as to what are the
roots of the polynomial, which motivates the definition of the affine hypersurface
of the polynomial.
Definition 1.1. Let F be a finite field with q a power of a prime p. The
q
n-dimensional affine space over the finite field F is the set of ordered n-tuples
q
(a ,...,a ) where each element a is in the finite field F . Denote the affine space
1 n i q
as An .
F
q
Let f(x ,...,x ) be a polynomial with n variables over F . Then the affine
1 n q
hypersurface H of the polynomial f(x ,...,x ) is the set of all the roots of
f 1 n
f(x ,...,x ) in the affine space An .
1 n F
q
H ={(a ,...,a )∈An |f(a ,...,a )=0}
f 1 n F 1 n
q
Let s be a positive integer and let F be a field extension of F . Observe that
qs q
thecoefficientsofthepolynomialf(x ,...,x )arealsoinF foranys. Thisallows
1 n qs
1
2 SUNWOOPARK
us to consider the set of all the roots of the polynomial f(x ,...,x ) in the affine
1 n
space of F .
qs
Definition 1.2. Let F be a field extension of F . Let H be the affine
qs q f
hypersurface of a polynomial f(x ,...,x ) with n variables over F . Then the set
1 n q
of F -points of H , denoted as H (F ), is the set of all the roots of f(x ,...,x )
qs f f qs 1 n
in the affine space of F .
qs
H (F )={(a ,...,a )∈An |f(a ,...,a )=0}
f qs 1 n Fqs 1 n
The order of the set H (F ) can be understood as the number of roots of the
f qs
polynomial f(x ,...,x ) over F . Using the definition above, we can construct a
1 n qs
function over C , the analogue of C in the field of p-adic numbers Q . Before we
p p
construct the function, we state the definition of the field of p-adic numbers Q .
p
Definition 1.3. Let p be a prime number. The ring of p-adic integers Z is the
p
inverse limitof theinversesystem ((Z/pnZ)n∈N,(fnm)n>m∈N)where thetransition
morphismf :Z/pnZ→Z/pmZisgivenbyreductionmodpm. Thefractionfield
nm
of Z is the field of p-adic numbers Q and the completion of the algebraic closure
p p
of Q is the complex field of p-adic numbers C .
p p
NoticeC isdefinedasabovebecausethealgebraicclosureofQ isnotcomplete.
p p
The valuation over Q and C is defined as follows, the construction of which is
p p
described in Chapter 1 and 3 of Koblitz [1].
Definition 1.4. The valuation of an element a ∈ Q , denoted as ord a, is the
p p
largest integer power l of p such that pl divides a. The p-adic norm of a is defined
as follows.
(cid:40)
p−ordpa if a(cid:54)=0
|a| =
p
0 if a=0
The norm defined above is a non-Archimedean norm. Let b ∈ Q have the
p
irreducible polynomial f(x)=xn+a xn−1+...+a over Q . Then the norm of b
1 n p
is defined as |b| = |a |1/n. The norm on Q extends to C by defining
p n p p p
|x| =lim |x | where x∈C and {x } is a sequence in Q .
p n→∞ n p p n p
Definition 1.5. Let f(x ,...,x ) be a polynomial with n variables over F for
1 n q
q = pr a power of p. Let H be the affine hypersurface of f(x ,...,x ). Denote
f 1 n
N = |H (F )| as the number of F -points of H . Then the zeta function of H
s f qs qs f f
is defined as follows, where Exp is the exponential function in C .
p p
Z(H ,T)=Exp (cid:32)(cid:88)∞ NsTs(cid:33)
f p s
s=1
The zeta function contains all the information on the number of roots of a
polynomial over each field extension of the finite field F . Bernard Dwork proved
q
the following property of the zeta function of H in 1960, which is the primary
f
focus of the paper.
Theorem 1.6 (Dwork, 1960). Let H be the affine hypersurface of a polynomial f
f
with n variables over the finite field F . Then the zeta function of H is a quotient
q f
of two polynomials with coefficients in Q .
p
RATIONALITY OF ZETA FUNCTIONS OVER FINITE FIELDS 3
In fact, this theorem is one of the statements of the Weil conjectures proposed
by Andr´e Weil in 1949. The Weil conjectures explain the properties of the zeta
functions of algebraic varieties over finite fields. Weil claimed that the zeta
functions are quotients of two polynomials over Q and satisfy some forms of
p
functional equations. He further claimed that the roots of the zeta functions
appear in restricted places, an analogue of the Riemann hypothesis. As a
particular example, the reciprocal roots of the zeta functions F(s) = Z(H ,q−s)
f
of projective 1-dimensional hypersurfaces H are on the line Re(s) = 1/2.
f
Dwork’s theorem corresponds to the rationality statement of the Weil conjectures.
Alexander Grothendieck proved the statement on functional equations in 1965
and Pierre Deligne proved the analogue of the Riemann hypothesis in 1973.
Dwork’s proof consists of two steps. The first step, proved by induction on
the number of variables, shows that the zeta function is a quotient of two power
series over C , both of which converge everywhere. Observe that N is the sum
p s
of the number of roots of the polynomial where each coordinate is non-zero and
the number of roots of the polynomial where at least one coordinate is zero. The
latter case inductively relates N with the number of roots of other polynomials
s
with fewer variables. The former case can be expressed as a sum of a set of p-th
roots of unity. By extending the coefficients of the polynomial from the finite field
F to the field of p-adic numbers Q , Dwork constructs a power series over C and
q p p
uses the series to change the sum of p-th roots of unity to the sum of power series
over C . He then shows that the determinant of the power series under the basis
p
of monomials converges everywhere on C , which proves the first step.
p
The second step uses the first step to prove the theorem. Using p-adic analysis,
DworkshowsthatthereexistsapolynomialandapowerseriesoverC withcertain
p
radiusofconvergencesuchthatthequotientofthepolynomialandthepowerseries
isthezetafunction. Hethencomparesthecoefficientsofthequotientwiththoseof
thezetafunction,whichallowshimtoestimatethep-adicnormandtheusualnorm
of the determinant of a matrix consisting of a finite set of coefficients of the zeta
function. The comparison between the two norms shows that the determinant of
thematrixis0whichimpliesthatthezetafunctionisaquotientoftwopolynomials
over C .
p
This paper focuses on carefully following Koblitz’s treatment [2] of the proof of
Dwork’s theorem [1]. The paper first provides important backgrounds on
properties of some power series over C . Using the properties, the paper then
p
follows Dwork’s proof on the rationality of zeta functions. The paper also states
an alternate construction of the zeta functions given by Kapranov [5], extending
the construction of the zeta functions over finite fields to zeta functions over
perfect fields.
2. Power Series over C
p
Let F(x) be a power series over C of the form F(x) = (cid:80) a xn. Since C
p n=0 n p
is complete and the norm is non-Archimedean, the series converges if and only if
the norm of the term |a xn| →0 as n→∞. In other words, for certain values of
n p
x ∈ C we can evaluate F(x) by the limit the infinite sum converges to. Observe
p
that if all the coefficients a are in Z , then the series F(x) converges for |x| <1
n p p
because |a xn| ≤|x|n →0 as n→∞. This proves the following lemma.
n p p
Lemma 2.1. If F(x) is a power series in Z , then F(x) converges for |x| <1.
p p
4 SUNWOOPARK
SimilartopowerseriesoverC,wecandefinetheradiusofconvergenceofapower
series over C .
p
Definition 2.2. The radius of convergence r of a power series F(x)=(cid:80)∞ a xn
n=0 n
is the following limit.
1
r =
limsup |a |1/n
n→∞ n p
As the term suggests, the series converges if |x| < r and diverges if |x| > r.
p p
This can be shown by observing whether the norm of the term a x approaches 0
n n
as n→∞. It is not true, however, that the series converges if |x| =r.
p
Definition 2.3. The closed and open disks of radius r ∈R around a point a∈C
p
are defined respectively as follows.
D (r)={x∈C ||x−a| ≤r}
a p p
D (r−)={x∈C ||x−a| <r}
a p p
WewillabbreviatethediskaroundtheoriginD (r)asD(r). Usingthisnotation,
0
we can say that a power series F(x) converges in D(r) or D(r−) if it has r as the
radius of convergence.
2.1. Logarithms, Exponents, and Other Power Series. Several power series
defined over C can be analogously defined over C .
p
Definition 2.4. ThelogarithmicfunctionLog andtheexponentialfunctionExp
p p
is defined as the following power series over C .
p
(cid:88)∞ xn
Log (1+x)= (−1)n+1
p n
n=1
(cid:88)∞ xn
Exp (x)=
p n!
n=0
As in the case for complex numbers, the following proposition holds.
Proposition 2.5. Let Log and Exp be the logarithmic and the exponential
p p
function in C . Then the following holds.
p
(1) Log (1+x) converges in D(1−).
p
(2) Expp(x) converges in D(p−p−11−).
(3) Log and Exp are mutually inverse isomorphisms between the
p p
multiplicative group D1(p−p−11−) and the additive group D(p−p−11−).
Proof. Observe that limsupn→∞|1/n|1p/n = limn→∞pordnpn = 1. Since |xn/n|p =
pordpn ≥1 if |x|p =1, Logp(1+x) converges in D(1−).
Noticethatordpn!= np−−S1n whereSn isthesumofallp-adicdigitsofn!. Suppose
that n has the p-adic expansion n = a pk +a pk−1 +...+a . Observe that
k k−1 0
ord n! = (cid:80)k (cid:98)n(cid:99) as (cid:98)n(cid:99) contributes to a factor p, (cid:98)n(cid:99) contributes to a square
p i=1 pi p p2
factorofp,andsoon. Thenforeveryiwehave(cid:98)n(cid:99)=a pk−i+a xk−1−i+...+a ,
pi k k−1 i
which implies the following.
ord n!=(cid:88)k (cid:98)n(cid:99)=(cid:88)k a pi−1 = (cid:80)ki=0(aipi)−(cid:80)ki=0ai = n−Sn
p pi i p−1 p−1 p−1
i=1 i=1
RATIONALITY OF ZETA FUNCTIONS OVER FINITE FIELDS 5
ThisshowsthattheradiusofconvergenceofExpp isp−p−11 asfromtheequation
below.
limsup|1/n!|1p/n =limsuppnn(−p−Sn1) =pp−11
n→∞ n→∞
−1
Suppose |x|p = pp−1. Choose n to be a power of p such that Sn = 1. Then
ordpxn/n! = pp−m1 − ppm−−11 = p−11, which shows |xn/n!|p = p−p−11. Hence Expp
converges in D(p−p−11−).
From the definition of Log , Log (1+x)(1+y) = Log (1+x)+Log (1+y),
p p p p
which shows that Log is a group homomorphism from the multiplicative group
p
D1(p−p−11−) to the additive group D(p−p−11−). Similarly, Expp is a group
homomorphism from D(p−p−11−) to D1(p−p−11−). By the definition of two
functions, we also have Exp (Log (1 + x)) = 1 + x and Log (Exp (x)) = x if
p p p p
x∈D(p−p−11−). Hence the two functions are mutually inverse isomorphisms. (cid:3)
Definition 2.6. Let a ∈ C . Then the binomial expansion (1+x)a is defined as
p
the following power series.
∞
(cid:88) a(a−1)...(a−n+1)
B (x)= xn
a n!
n=0
If|a| >1thenforanyintegeri,|a−i| =|a| . Hencethenormofthenthterm
p p p
ofthebinomialpowerseriesis|anxn| . Thisimpliesthatthepowerseriesconverges
n! p
in D(p−p−11−). If |a| ≤1 then for any integer i, |a−i| ≤1. This shows that the
|a|p p p
norm of the nth term of the binomial power series is |a(a−1)...(a−n+1)xn| ≤ |xn| .
n! p n! p
This implies that the power series converges in D(p−p−11−).
We now define a new power series which will be used in Dwork’s proof of the
rationality of zeta functions.
Definition 2.7. Let F(x,y) be a power series with two variables over C defined
p
as follows.
∞
F(x,y)=B (y)(cid:89)B (ypi+1)
x xpi+1−xpi
i=0 pi+1
This can be rewritten as follows.
F(x,y)=(1+y)x(1+yp)xpp−x...(1+ypn)xpn−pxnpn−1...
Observe that the power series F(x,y) has all its coefficients in Q . The series
p
is also well defined because for any term xnym, the corresponding coefficient can
be achieved by taking finitely many terms in Q . We will analyze this power series
p
further after proving the following lemma.
Lemma 2.8 (Dwork’s Lemma). Let F(x) be a power series over Q such that the
p
constant term is 1. Then F(x) has all its coefficients in Z if and only if F(xp)
p (F(x))p
has all its coefficients other than the constant term in pZ .
p
Proof. SupposeF(x)hasallitscoefficientsinZ withconstantterm1. Noticethat
p
(a+b)p ≡ ap+bp mod p and ap ≡ a mod p for any a,b ∈ Z . Then there exists a
p
6 SUNWOOPARK
power series G(x)∈xZ [[x]] such that the following holds.
p
(F(x))p =F(xp)+pG(x)
The above equation implies the following equation because (F(x))p has all its
coefficients in Z and 1+xZ [[x]] is a group under multiplication.
p p
F(xp) G(x)
=1−p ∈1+pxZ [[x]]
(F(x))p (F(x))p p
Suppose F(xp) hasallitscoefficientsotherthantheconstantterminpZ . Then
(F(x))p p
we can find a power series G(x) ∈ 1+pxZ [[x]] such that F(xp) = (F(x))pG(x).
p
Let F(x)=(cid:80)∞ a xi and G(x)=(cid:80)∞ b xj where b ∈pZ for all j. We want to
i=0 i j=0 j j p
show that a ∈Z for all i.
i p
We prove the lemma by induction on the index i. The base case holds because
a =1. Suppose a ∈Z for i≤n−1. Consider the following equation.
0 i p
F(xp)=(F(x))pG(x)
The coefficients of xn on both sides of the equation must be the same. Notice
that the coefficient of xn from the RHS of the equation above is determined by
the product ((cid:80)n a xi)p(1+(cid:80)n b xj). This shows that the coefficient from the
i=0 i j=1 j
RHS consists of terms which contain b for some j and terms which only have a
j i
as factors. The former terms are in pZ , which we will denote as pB.
p
If p | n, then the coefficient from the LHS is a while the coefficient from the
n/p
RHS is pB+pa +ap . For any a ∈ Z we have ap ≡ a mod p. Hence a is an
n n/p p n
element of Z . If p (cid:45) n then the coefficient from the LHS is 0 while the coefficient
p
from the RHS is pB+pa . This also shows a is an element of Z . (cid:3)
n n p
ThegeneralizationofDwork’slemmaisasfollows,theproofofwhichisanalogous
to that of Lemma 2.8.
Lemma 2.9 (GeneralizedDwork’sLemma). Let F(x ,x ,...,x ) be a power series
1 2 n
with n variables over Q such that the constant term is 1. Then F(x ,...,x ) has
p 1 n
tahlleictsoncosteaffinctiteenrtms iinnZppZif.and only if (FF((xx1p1,,......,,xxnpn)))p has all its coefficients other than
p
Using the generalized Dwork’s lemma, we prove the following proposition.
Proposition 2.10. The power series F(x,y) has all its coefficients in Z .
p
Proof. Observe that the following equation holds by Lemma 2.9.
F(xp,yp) Bxp(yp)(cid:81)∞i=0B(xpi+2−xpi+1)/pi+1(ypi+2)
=
(F(x,y))p B (y)(cid:81)∞ B (ypi+1)
px i=0 (xpi+1−xpi)/pi
= Bxp(yp) (cid:81)∞i=0B(xpi+2−xpi+1)/pi+1(ypi+2)
B (y)B (yp) (cid:81)∞ B (ypi+1)
px xp−x i=1 (xpi+1−xpi)/pi
(1+yp)xp (1+yp)x
= =
(1+y)px(1+y)xp−x (1+y)px
Henceitsufficestoshowthat (1+yp)x hasallitscoefficientsotherthantheconstant
(1+y)px
term in pZ . Notice that 1+y is an element of 1+yZ [[y]]. By Lemma 2.8, there
p p
RATIONALITY OF ZETA FUNCTIONS OVER FINITE FIELDS 7
exists a power series G(y)∈Z [[y]] such that the following holds.
p
(1+yp)
=1+pyG(y)
(1+y)p
By the definition of binomial power series, the following equation holds.
(1+yp)x
=(1+pyG(y))x ∈1+pxZ [[x,y]]+pyZ [[x,y]]
(1+y)px p p
(cid:3)
The proposition shows that F(x,y) can be expressed as follows in which for all
non-negative integers n and m, a ∈Z .
m,n p
∞ ∞
(cid:88) (cid:88)
F(x,y)= (xn a ym)
m,n
n=0 m=n
Notice we have (cid:80)∞ a ym instead of (cid:80)∞ a ym because for each term in
m=n m,n m=0 m,n
B (y) the power of x is less than or equal to the power of y, i.e.
xpi+1−xpi
pi+1
(cid:32) (cid:33)(cid:32) (cid:33) (cid:32) (cid:33)
xpi+1 −xpi xpi+1 −xpi xpi+1 −xpi ynpi
−1 ... −n+1
pi+1 pi+1 pi+1 n!
2.2. Newton Polygons.
Definition 2.11. Let f(x)=1+(cid:80)n a xi be a polynomial of degree n over C .
i=1 i p
The Newton polygon of f(x) is the convex hull of the points (0,0), (1,ord a ), ...,
p 1
(i,ord a ), ..., (n,ord a ) in the real coordinate plane. In other words, it is the
p i p n
highest convex polygonal line which starts at (0,0), ends at (n,ord a ), and joins
p n
or passes below all the points (i,ord a ).
p i
Let F(x) = 1+(cid:80)∞ a xi be a power series over C . Let f (x) be the n-th
i=1 i p n
partial sum of F(x). Then the Newton polygon of F(x) is the limit of the Newton
polygons of f (x).
n
ThewaytoconstructtheNewtonpolygonofapolynomialorapowerseriesisas
follows. Plotallthesetofpoints(i,ord i). Rotatetheverticallinepassingthrough
p
(0,0) counter-clockwise until it hits a point (i,ord a ) for some i. Then rotate the
p i
line with respect to the point (i,ord a ) and repeat the process.
p i
For convenience we will define the terms as follows. The set of vertices of the
Newton polygon is the set of points where the slopes change. The length of a
segment is the length of the projection of the segments onto the horizontal axis.
The Newton polygon of a polynomial f(x) over C gives important properties
p
of the roots of f(x). We omit the proof of the following lemma which is written in
Chapter 4 of Koblitz [2].
Lemma 2.12. Let f(x) = (cid:81)n (1 − x/a ) be a polynomial over C with roots
i=1 i p
{a }n . Let ord a = m . If m is a slope of the Newton polygon of f(x) with
i i=1 p i i
length q, then precisely q of m are equal to m.
i
The Newton polygon of a power series can be classified into three categories.
(1) There are infinitely many segments of finite length. Consider the power
series F(x) = 1+(cid:80)∞ pi2xi. The Newton polygon is an infinite set of
n=1
finitelinesegmentswhichconnectsthelattices(n,n2)onthegraphy =x2.
8 SUNWOOPARK
(2) Therearefinitelymanysegmentinwhichthelastsegmentisinfinitelylong.
The power series F(x)=1+(cid:80)∞ xn has an infinite horizontal line as its
n=1
Newton polygon.
(3) TheremaybeacasesuchthatitisnotpossibletodrawtheNewtonpolygon
using the method above. Consider the power series F(x)=1+(cid:80)∞ pxn.
n=1
Observe that there exists an integer n such that the point (n,1) lies below
a line segment which is obtained from rotating the horizontal line passing
through the origin counterclockwise. In this case, we let the last segment
of the Newton polygon have the slope to be the least upper bound of all
slopesofwhichanypoint(i,ord a )liesaboveorontheline. Intheprevious
p i
example,theNewtonpolygonistheinfinitehorizontallinepassingthrough
the origin.
Newton polygons provide a different way of understanding the power series over
C . TheslopeoftheNewtonpolygonofapowerseriesgivesadditionalinformation
p
on the radius of convergence of the series as well as the valuations of the roots
of the power series. The following theorem, which is an analogue of Lemma 2.12
for power series, will prove useful for understanding Dwork’s proof in subsequent
sections.
Theorem 2.13 (p-adic Weierstrass Preparation Theorem). Let F(x) be a power
series over C which converges on D(pm). Denote the terms of the power series as
p
F(x) = 1+(cid:80) a xn . Suppose the Newton polygon of F(x) does not have the
n=1 n
infinitely long last segment of slope m. Denote N as the total length of all finite
segments having slope less than m. If the Newton polygon of F(x) has the last
segment of slope m, denote N as the greatest n such that (n,ord a ) lies on the
p n
last segment. Then there exists a unique polynomial h(x) in C with constant term
p
1 of degree N and a power series G(x) = 1+(cid:80)∞ b xn which converges and is
n=1 n
nonzero in D(pm) such that h(x)=F(x)G(x).
Corollary 2.14. Suppose F(x) is a power series over C with the constant term
p
1. If a segment of the Newton polygon of F(x) has horizontal length N and slope
m, then there are precisely N roots x of F(x) counting multiplicity such that the
valuation ord x=−m.
p
We first prove the following lemmas which will help us prove Theorem 2.13.
Lemma 2.15. Let F(x) = 1+(cid:80)∞ a xn be a power series over C . Let m be
n=1 n p
the least upper bound of all the slopes of the Newton polygon of F(x). Here m may
be infinite. Then the radius of convergence of F(x) is pm.
Proof. Suppose x ∈ C such that ord x = −m(cid:48) > −m. Then ord (a xn) =
p p p n
ord a −nm(cid:48). Noticethat(n,ord a )liesabovetheNewtonpolygonwhile(n,nm(cid:48))
p n p n
lies above a line of slope m(cid:48) passing through the origin. By the definition of m,
ord a −nm(cid:48) →∞asn→∞,whichimpliesthattheseriesconverges. Nowassume
p n
ord x=−m(cid:48)(cid:48) <−m. By the similar argument as above, for infinitely many n, the
p
value ord a −nm(cid:48)(cid:48) is negative, proving that the series does not converge. (cid:3)
p n
The above lemma does not explain whether the power series converges when
ord x is equal to the least upper bound of all slopes of the Newton polygon.
p
Remark 2.16. Let F(x) = 1+(cid:80)∞ a xn be a power series over C . Suppose
n=1 n p
c ∈ C such that ord c = d and G(x) = F(x/c) = 1+(cid:80)∞ b xn is also a power
p p n=1 n
RATIONALITY OF ZETA FUNCTIONS OVER FINITE FIELDS 9
series over C . Notice that for each n, ord b =ord (a /cn)=ord a −dn. This
p p n p n p n
shows that the Newton polygon of G(x) is obtained by subtracting the line y =dx
from the Newton polygon of F(x).
Lemma2.17. Supposem isthefirstslopeoftheNewtonpolygonofapowerseries
1
F(x)=1+(cid:80)∞ a xn over C . Let c∈C such that ord c=m≤m .
n=1 n p p p 1
Let G(x) be the power series defined as G(x) = (1−cx)F(x). Suppose F(x)
converges on the closed disk D(pm). Then the Newton polygon of G(x) is obtained
by first joining (0,0) to (1,m) and then shifting the Newton polygon of F(x) by 1
to the right and d upwards.
SupposefurtherthatthelastslopeoftheNewtonpolygonofF(x)ism . Thenthe
f
Newton polygon of G(x) also has m as the last slope. In addition, F(x) converges
f
in D(pmf) if and only if G(x) does.
Proof. We will first prove the lemma when c = 1. Then G(x) = (1 − x)F(x)
satisfies G(x) = 1+(cid:80)∞ (a −a )xn. Denote b = a −a . It follows that
n=1 n n−1 n n n−1
ord b ≥min(ord a ,ord a ). Since both points (n,ord a ),(n−1,ord a )
p n p n p n−1 p n p n−1
lieaboveorontheNewtonpolygonofF(x),thepoints(n,ord b )alllieabovethe
p n
Newton polygon of F(x) translated by 1 to the right. Notice that the translation
includes an extra segment connecting (0,0) and (1,0). If (n−1,ord a ) is one
p n−1
of the vertices of the Newton polygon of F(x), then ord b = ord a because
p n p n−1
ord a > ord a . This shows (n,ord b ) is also a corner of the translated
p n p n−1 p n
Newton Polygon.
Now we prove the second statement of the lemma. Notice that since ord b ≥
p n
min(ord a ,ord a ), G(x) converges whenever F(x) converges. Suppose that
p n p n−1
the Newton polygon of G(x) has a slope such that m > m . Then for some n,
g f
(n+1,ord a ) lies below the Newton polygon of G(x). This shows that for all
p n
k ≥ n+1, ord b > ord a . Since a = b +a , ord a = ord a . By
p k p n n+1 n+1 n p n+1 p n
the similar argument, ord a = ord a for all such k ≥ n+1. This contradicts
p k p n
the assumption that the power series F(x) converges on D(1). The converse holds
analogously.
Now we prove the lemma for arbitrary choice of c. Define the power series
F (x)=F(x/c) and G (x)=(1−x)F (x). Notice that the lemma holds for F (x)
1 1 1 1
andG (x)becausebothpowerseriessatisfythebasecaseoftheproofofthelemma.
1
ByRemark2.16,weobtainthedesiredresultsfortheNewtonpolygonofG(x). (cid:3)
Lemma 2.18. Let F(x) = 1+(cid:80)∞ a xn be a power series over C . Suppose
n=1 n p
the Newton polygon of F(x) has the first slope m . If F(x) converges on the closed
1
disk D(pm1) and the line through (0,0) with slope m1 passes through (n,ordpan)
for some n, then there exists a root x of F(x) such that ord x=−m .
p 1
Proof. We first prove the lemma when m =0. This implies that for all n we have
1
ord a ≥0andord a →∞asn→∞. LetM bethegreatestintegernsuchthat
p n p n
ord a = 0. Denote f (x) as the n-th partial sum of F(x). By Lemma 2.12, for
p n n
n ≥ M, the polynomial f (x) has precisely M roots x ,x ,...,x such that
n n,1 n,2 n,M
for every i, ord x =0.
p n,i
LetS bethesetofrootsoff (x)countingmultiplicities. Considerthefollowing
n n
sequence {x } such that x = x and x = a ∈ S such that ord a = 0
n M M,1 n+1 n+1 p
and |a−x | is minimal. Notice that if a ∈ S +1 such that ord a < 0, then
n n p
|1−x /a| =1. Hence the following holds for n>M. The equation below shows
n p
that {x } is a cauchy sequence because |a | →∞ as n→∞.
n n p
10 SUNWOOPARK
|a +1| =|a xn+1| =|f (x)−f (x)|
n p n+1 n p n+1 n p
=|f (x )| = (cid:89) (cid:12)(cid:12)1− xn(cid:12)(cid:12)
n+1 n p (cid:12) a (cid:12)p
a∈Sn+1
M (cid:12) (cid:12) M
=(cid:89)(cid:12)(cid:12)(cid:12)1− xxn (cid:12)(cid:12)(cid:12) =(cid:89)|xn+1,i−xn|p
i=1 n+1,i p i=1
≥|x −x |
n+1 n p
Suppose x is the limit of the sequence {x }. Then the following holds.
n
(cid:12) (cid:12)
(cid:12)(cid:88)n xk−xk(cid:12)
|f (x)| =|f (x)−f (x )| =|x−x | (cid:12) a n(cid:12)
n p n n n p n p(cid:12) k x−x (cid:12)
(cid:12) n (cid:12)
k=1 p
≤|x−x | →0asn→∞
n p
Since F(x) = lim f (x) = 0, x is the root of the power series, proving the
n→∞ n
lemma. Noticethatthegeneralcasefollowsdirectlyfromthebasecase. Letb∈C
p
such that ord b = m . Define G(x) = F(x/b). Then G(x) satisfies the conditions
p 1
for the base case of the proof. The analogous result holds for F(x) as well. (cid:3)
Lemma 2.19. Let F(x) = 1 + (cid:80)∞ a xn be a power series over C which
n=1 n p
converges and vanishes to 0 at a ∈ C . Let G(x) = F(x) = 1+(cid:80)∞ b xk be a
p 1−x/a k=1 k
power series over C . Then G(x) converges on D(|a| ).
p p
Proof. Letf (x)bethen-thpartialsumofF(x). NoticethatG(x)canbeobtained
n
bymultiplyingF(x)withthepowerseries(cid:80)∞ xm/am. Thisimpliesthatforeach
m=0
n, b = (cid:80)n aj = fn(a). Since f(a) = 0, we have |b an| = |f (a)| → 0 as
n j=0 an−j an n p n p
n→∞. (cid:3)
We now prove Theorem 2.13.
Theorem 2.13. We prove the theorem by induction on N, the degree of the
polynomial h(x). Without loss of generality, assume m = 0. The general case for
any value of m is analogous to the proof of Lemma 2.17 and Lemma 2.18.
Suppose N = 0. It suffices to show that the inverse power series of F(x) is
convergent and nonzero in D(pm). Let G(x) be the inverse of F(x). Assume for
everyn,ord a >0. Observethatord a →∞asn→∞becauseF(x)converges
p n p n
in D(1). Since F(x)G(x) = 1, b = −(cid:80)n b a , which shows that ord b > 0
n k=1 n−k k p n
for every n. Notice that ord b → ∞ as n → ∞. For fixed C > 0, choose c
p n
such that for all n > c, ord a > C. Denote (cid:15) = min(ord a ,ord a ,...,ord a ).
p n p 1 p 2 p c
We claim that for all n > ic, ord b > min(C,i(cid:15)), which proves the theorem for
p n
N = 0. We prove the theorem by by induction on i. The case in which i = 0 is
trivial. Supposetheclaimholdsfori−1. Thenfromthesumb =−(cid:80)n b a ,
n k=1 n−k k
the terms b a with k > c have ord (b a ) ≥ ord a > C while the terms
n−k k p n−k k p k
b a with k ≤c have ord (b a )≥ord b +(cid:15)>min(C,(i−1)(cid:15))+(cid:15) by the
n−k k p n−k k p n−k
induction hypothesis. Hence ord b >min(C,i(cid:15)).
p n
Now suppose N ≥1 and the theorem holds for N −1. Let m ≤m be the first
1
slope of the Newton polygon of F(x). By Lemma 2.18, there exists a ∈ C such
p
that F(a)=0 and ord a=−m . Consider the following power series over C .
p 1 p
