Computations with p-adic numbers

0.81 MB·

by  Xavier Caruso

Computations with p-adic numbers
Xavier Caruso
January 25, 2017
Abstract
7 This document contains the notes of a lecture I gave at the “Journ´ees Nationales du
1 Calcul Formel1” (JNCF) on January 2017. The aim of the lecture was to discuss low-level
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algorithmics forp-adic numbers. It is divided into two main parts: first,we present various
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implementationsof p-adic numbersand compare them and second, we introduce ageneral

n
frameworkforstudyingprecisionissuesandapplyitinseveralconcretesituations.
a
J

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Contents
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] 1 Introduction to p-adic numbers 3
T
N 1.1 Definition andfirst properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
. 1.2 Newton iteration overthe p-adic numbers . . . . . . . . . . . . . . . . . . . . . . 9
h
1.3 Similarities with formalandLaurentseries . . . . . . . . . . . . . . . . . . . . . . 13
t
a
1.4 Why should we implementp-adic numbers? . . . . . . . . . . . . . . . . . . . . . 15
m
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2 Several implementations of p-adic numbers 19

1 2.1 Zealousarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
v
2.2 Lazy and relaxedarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4
9 2.3 Floating-pointarithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7 2.4 Comparison betweenparadigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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3 The art of tracking p-adic precision 42
.
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3.1 Foundations ofthe theory of p-adic precision . . . . . . . . . . . . . . . . . . . . . 43
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7 3.2 Optimal precision and stability ofalgorithms . . . . . . . . . . . . . . . . . . . . . 53
1 3.3 Lattice-based methods for trackingprecision . . . . . . . . . . . . . . . . . . . . . 70
:
v
i References 81
X
r
a
Introduction
The field of p-adic numbers, Q , was first introduced by Kurt Hensel at the end of the 19th
p
century in a short paper written in German [36]. From that time, the popularity of p-adic
numbers has grown without interruption throughout the 20th century. Their first success was
materialized by the famous Hasse–Minkowski’s Theorem [75] that states that a Diophantine
equation of the form P(x ,...,x ) = 0 where P is a polynomial of total degree at most 2 has
1 n
a solution over Q if and only if it has a solution over R and a solution over Q for all prime
p
numbers p. This characterization is quite interesting because testing whether a polynomial
equation has a p-adic solution can be carried out in a very efficient way using analytic methods
just like over the reals. This kind of strategy is nowadays ubiquitous in many areas of Number
1(French)NationalComputerAlgebraDays
1
Theory and Arithmetic Geometry. After Diophantine equations, other typical examples come
from the study of number fields: we hope deriving interesting information about a number
field K by studying carefully all its p-adic incarnations K Q . The ramification of K, its
Q p
⊗
Galois properties, etc. can be — and are very often — studied in this manner [69, 65]. The
class field theory, which provides a precise description of all Abelian extensions2 of a given
number field, is also formulated in this language [66]. The importance of p-adic numbers is so
prominenttodaythatthereisstillnowadaysveryactiveresearchontheorieswhicharededicated
to purely p-adic objects: one can mention for instance the study of p-adic geometry and p-adic
cohomologies [6, 58], the theory of p-adic differential equations [50], Coleman’s theory of p-
adic integration [24], the p-adic Hodge theory [14], the p-adic Langlands correspondence [5],
the study of p-adic modular forms [34], p-adic ζ-functions [52] and L-functions [22], etc. The
proofofFermat’slastTheorembyWilesandTaylor[81,78]isstampedwithmanyoftheseideas
anddevelopments.
Over the last decades, p-adic methods have taken some importance in Symbolic Computa-
tion as well. For a long time, p-adic methods have been used for factoring polynomials over
Q [56]. More recently, there has been a wide diversification of the use of p-adic numbers for
effective computations: Bostan et al. [13] used Newton sums for polynomials over Z to com-
p
putecomposedproductsforpolynomialsoverF ;Gaudryetal.[32]usedp-adicliftingmethods
p
to generate genus 2 CM hyperelliptic curves; Kedlaya [49], Lauder [54] and many followers
usedp-adiccohomologytocountpointsonhyperellipticcurvesoverfinitefields;LercierandSir-
vent [57] computed isogenies between elliptic curves over finite fields using p-adic differential
equations.
Theneedtobuildsolidfoundationstothealgorithmicsofp-adicnumbershasthenemerged.
This is however not straightforward because a single p-adic number encompasses an infinite
amountofinformation(theinfinitesequenceofitsdigits) andthennecessarilyneedstobetrun-
cated in order to fit in the memory of a computer. From this point of view, p-adic numbers
behave very similarly to real numbersand the questions that emerge when we are trying to im-
plementp-adicnumbersareoftenthesameasthequestionsarisingwhendealingwithrounding
errors in the real setting [62, 26, 63]. The algorithmic study of p-adic numbers is then located
at the frontier between Symbolic Computation and Numerical Analysis and imports ideas and
resultscomingfrom bothof these domains.
Contentand organizationof this course. This course focuses on the low-levelimplementation of
p-adicnumbers(andthenvoluntarilyomitshigh-levelalgorithmsmakinguseofp-adicnumbers)
and pursues two main objectives. The first one is to introduce and discuss the most standard
strategiesforimplementingp-adicnumbersoncomputers. Weshalldetailthreeofthem,eachof
them having its own spirit: (1)the zealousarithmetic which is inspired by interval arithmetic in
the real setting, (2) the lazy arithmetic with its relaxed improvementand (3) the p-adicfloating-
pointarithmetic, the lasttwo being inspired by the eponym approachesin the realsetting.
The second aim of this course is to develop a general theory giving quite powerful tools to
study the propagation of accuracy in the p-adic world. The basic underlying idea is to linearize
the situation (and then model the propagation of accuracy using differentials); it is once again
inspiredfromclassicalmethodsintherealcase. However,itturnsoutthatthenon-archimedean
nature of Q (i.e. the fact that Z is bounded in Q ) is the source of many simplifications which
p p
will allow us to state much more accurate results and to go much further in the p-adic setting.
As an example, we shall see that the theory of p-adic precision yields a general strategy for
increasingthenumericalstability ofanygivenalgorithm(assumingthattheproblemitsolvesis
well-conditioned).
This course is organized as follows. 1 is devoted to the introduction of p-adic numbers:
§
we define them, prove their main properties and discuss in more details their place in Num-
2AnabelianextensionisaGaloisextensionwhoseGaloisgroupisabelian.
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ber Theory, Arithmetic Geometry and Symbolic Computation. The presentation of the standard
implementations of p-adic numbers mentioned above is achieved in 2. A careful comparison
§
between them is moreover proposed and supported by many examples coming from linear al-
gebra and commutative algebra. Finally, in 3, we expose the aforementioned theory of p-adic
§
precision. We then detail its applications: we will notably examine many very concrete situa-
tionsand,foreachofthem,wewillexplainhowthetheoryofp-adicprecisionhelpsuseitherin
quantifying the qualities ofa given algorithm regardingto numericalstability or,even better,in
improvingthem.
Acknowledgments. This document contains the (augmented) notes of a lecture I gave at the
“Journ´eesNationalesduCalculFormel3”(JNCF)onJanuary2017. Iheartilythanktheorganizers
and the scientific committee of the JNCF for giving me the opportunity to give these lectures
and for encouraging me to write down these notes. I am very grateful to Delphine Boucher,
Nicolas Brisebarre, Claude-Pierre Jeannerod, Marc Mezzarobba and Tristan Vaccon for their
carefulreading andtheir helpful commentson an earlierversion ofthese notes.
Notation. We use standard notation for the set of numbers: N is the set of natural integers
(including 0), Z is the set of relative integers, Q is the set of rational numbers and R is the set
of real numbers. We will sometimes use the soft-O notation O˜( ) for writing complexities; we
−
recallthat, given a sequenceof positive real numbers(u ), O˜(u ) is definedas the union of the
n n
sets O(u logku ) for k varying in N.
n n
Throughoutthis course,the letterp always refersto afixed primenumber.
1 Introduction to p-adic numbers
In this first section, we define p-adic numbers, discuss their basic properties and try to explain,
by selecting a few relevant examples, their place in Number Theory, Algebraic Geometry and
Symbolic Computation. The presentation below is voluntarily very summarized; we refer the
interestedreaderto [2,35]for a morecompleteexposition ofthe theory ofp-adic numbers.
1.1 Definition and first properties
p-adic numbers are very ambivalent objects which can be thought of under many different an-
gles: computational, algebraic, analytic. It turns out that each point of view leads to its own
definition of p-adic numbers: computer scientists often prefer viewing a p-adic number as a
sequence of digits while algebraists prefer speaking of projective limits and analysts are more
comfortable with Banach spaces and completions. Of course all these approaches have their
own interest and understanding the intersections between them is often the key behind the
mostimportant advances.
In this subsection, we briefly survey all the standard definitions of p-adic numbers and pro-
vide several mental representations in order to try as much as possible to help the reader to
developagood p-adic intuition.
1.1.1 Down-to-earth definition
Recallthat eachpositive integer n canbe written in basep,that is as afinite sum:
n= a +a p+a p2+ +a pℓ
0 1 2 ℓ
···
where the a ’s are integers between 0 and p 1, the so-called digits. This writing is moreover
i
−
unique assuming that the most significant digit a does not vanish. A possible strategy to com-
ℓ
putethe expansioninbase pgoesasfollows. We firstcomputea bynotingthat itisnecessarily
0
3(French)NationalComputerAlgebraDays
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a : 1742 = 248 7 + 6
0
×
a : 248 = 35 7 + 3
1
×
a : 35 = 5 7 + 0
2
×
a : 5 = 0 7 + 5
3
×
7
= 1742 = 5036
⇒
Figure1.1: Expansion of1742 in base 7
... 2 3 0 6 2 4 4 ... 2 3 0 6 2 4 4
+ ... 1 6 5 2 3 3 2 ... 1 6 5 2 3 3 2
×
... 4 2 6 1 6 0 6 ... 4 6 1 5 5 2 1
... 2 2 5 0 6 5
... 2 5 0 6 5
... 5 5 2 1
... 6 1 6
... 6 3
... 4
... 4 3 2 0 3 0 1
Figure1.2: Addition andmultiplication in Z
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the remainderin the Euclidean division ofn by p: indeedit is congruentto n modulo p andlies
in the range [0,p 1] by definition. Once a is known, we compute n = n−a0, which is also the
− 0 1 p
quotientintheEuclideandivisionofnbyp. Clearlyn = a +a p+ +a pℓ−1 andwecannow
1 1 2 ℓ
···
computea repeatingthe same strategy. Figure1.1 shows a simple executionofthis algorithm.
1
By definition, a p-adicintegeris an infinite formalsum of the shape:
x = a +a p+a p2+ +a pi+
0 1 2 i
··· ···
where the a ’s are integers between 0 and p 1. In other words, a p-adic integer is an integer
i
−
written in base p with an infinite number of digits. We will sometimes alternatively write x as
follows:
x = ...a ...a a a a p
i 3 2 1 0
or simply
x = ...a ...a a a a
i 3 2 1 0
when no confusion can arise. The set of p-adic integers is denoted by Z . It is endowed with a
p
naturalstructureofcommutativering. Indeed,wecanadd,subtractandmultiplyp-adicintegers
usingtheschoolbook method;note thathandlingcarriesispossible sincethey propagateonthe
left. The ring of natural integers N appears naturally as a subring of Z : it consists of p-adic
p
integers ...a ...a a a a for which a = 0 when i is large enough. Note in particular that the
i 3 2 1 0 i
integer p writes ...0010 in Z and more generally pn writes ...0010...0 with n ending zeros.
p
Asaconsequence,ap-adicintegerisamultipleofpn ifandonlyifitendswith(atleast)nzeros.
Remarkthatnegativeintegersare p-adicintegersaswell: the opposite ofnis, by definition, the
resultof the subtraction 0 n.
−
Similarly,we define ap-adic numberasa formalinfinite sum ofthe shape:
x = a p−n+a p−n+1+ +a pi+
−n −n+1 i
··· ···
where nis an integerwhich may dependon x. Alternatively,we will write:
x = ...a ...a a a .a a ...a p
i 2 1 0 −1 −2 −n
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and, when no confusion may arise, we will freely remove the bar and the trailing p. A p-adic
number is then nothing but a “decimal” number written in base p with an infinite number of
digits before the decimal mark and a finite amount of digits after the decimal mark. Addition
andmultiplication extendto p-adic numbersas well.
The set of p-adic numbers is denoted by Q . Clearly Q = Z [1]. We shall see later (cf
p p p p
Proposition 1.1, page 5) that Q is actually the fraction field of Z ; in particular it is a field and
p p
Q,which is the fraction fieldofZ,naturallyembedsinto Q .
p
1.1.2 Second definition: projective limits
From the point of view of addition and multiplication, the last digit of a p-adic integer behaves
like an integer modulo p, that is an element of the finite field F = Z/pZ. In other words, the
p
application π : Z Z/pZ taking a p-adic integer x = a +a p+a p2 + to the class of a
1 p 0 1 2 0
→ ···
modulop is a ringhomomorphism. Moregenerally,given apositive integern,the map:
π : Z Z/pnZ
n p
→
a +a p+a p2+ (a +a p+ +a pn−1) modpn
0 1 2 0 1 n−1
··· 7→ ···
isaringhomomorphism. Thesemorphismsarecompatibleinthefollowingsense: forallx Z ,
p
∈
wehaveπ (x) π (x) (mod pn)(andmoregenerallyπ (x) π (x) (mod pn)providedthat
n+1 n m n
≡ ≡
m > n). Putting the π ’s all together,we endupwith aring homomorphism:
n
π : Z lim Z/pnZ
p
→ n
←−
x (π (x),π (x),...)
1 2
7→
wherelim Z/pnZisbydefinitionthesubringof ∞ Z/pnZconsistingofsequences(x ,x ,...)
n n=1 1 2
for whi←c−h xn+1 xn (mod pn) for alln: it is calQledthe projective limit of the Z/pnZ’s.
≡
Conversely, consider a sequence (x ,x ,...) lim Z/pnZ. In a slight abuse of notation,
1 2
∈ n
continue to write x for the unique integer of the←r−ange J0,pn 1K which is congruent to x
n n
−
modulopn and write it in base p:
x = a +a p+ +a pn−1
n n,0 n,1 n,n−1
···
(the expansion stops at (n 1) since x < pn by construction). The condition x x
n n+1 n
− ≡
(mod pn) implies that a = a for all i J0,n 1K. In other words, when i remains fixed,
n+1,i n,i
∈ −
the sequence(a ) is constant andthus convergesto some a . Set:
n,i n>i i
ψ(x ,x ,...) = ...a ...a a a Z .
1 2 i 2 1 0 p
∈
We define this way an application ψ : lim Z/pnZ Z which is by construction a left and a
p
n →
rightinverse ofπ. In other words, π and←−ψ are isomorphisms whichare inversesof eachother.
The above discussion allowsusto give an alternativedefinition of Z , which is:
p
Z =lim Z/pnZ.
p
←n−
The map π then corresponds to the projection onto the n-th factor. This definition is more
n
abstract and it seems more difficult to handle as well. However it has the enormousadvantage
of making the ring structure appear clearly and, for this reason, it is often much more useful
andpowerfulthan thedown-to-earth definition of 1.1.1. Asatypical example,letusprovethe
§
followingproposition.
Proposition 1.1. (a) An elementx Z is invertible in Z if andonlyif π (x) doesnotvanish.
p p 1
∈
(b) The ring Q is the fractionfieldof Z ; in particular, it is a field.
p p
5
Proof. (a)Let x Z . Viewing Z as lim Z/pnZ,we findthat x is invertiblein Z if andonlyif
p p p
∈ n
π (x) is invertible in Z/pnZ for all n.←−The latest condition is equivalent to requiring that π (x)
n n
and pn are coprime for all n. Noting that p is prime, this is further equivalent to the fact that
π (x) mod p = π (x) doesnot vanish in Z/pZ.
n 1
(b) By definition Q = Z [1]. It is then enough to prove that any nonzero p-adic integer x can
p p p
be written as a product x = pnu where n is a nonnegative integer and u is a unit in Z . Let
p
n be the number of zeros at the end of the p-adic expansion of x (or, equivalently, the largest
integernsuchthatπ (x) = 0). Thenxcanbewrittenpnuwhereuisap-adicintegerwhoselast
n
digit does not vanish. By the first part of the proposition, u is then invertible in Z and we are
p
done.
We note that the first statement of Proposition 1.1 shows that the subset of non-invertible
elements of Z is exactly the kernel of π . We deduce from this that Z is a local ring with
p 1 p
maximalideal kerπ .
1
1.1.3 Valuation and norm
We define the p-adic valuationof the nonzerop-adic number
x =...a ...a a a .a a ...a
i 2 1 0 −1 −2 −n
as the smallest (possibly negative) integer v for which a does not vanish. We denote it val (x)
v p
or simply val(x) if no confusion may arise. Alternatively val(x) can be defined as the largest
integer v such that x pvZ . When x = 0, we put val(0) = + . We define this way a function
p
∈ ∞
val : Q Z + . Writing down the computations (and remembering that p is prime), we
p
→ ∪{ ∞}
immediately checkthe followingcompatibility properties for all x,y Q :
p
∈
(1) val(x+y) > min val(x),val(y) ,
(cid:0) (cid:1)
(2) val(xy) = val(x)+val(y).
Note moreoverthat the equality val(x+y) = min val(x),val(y) does hold as soon as val(x) =
6
val(y). As we shall see later,this property reflectst(cid:0)he tree structu(cid:1)reof Z (see 1.1.5).
p
§
Thep-adicnorm isdefinedby x =p−val(x) forx Q . Inthesequel,whennoconfusion
p p p
|·| | | ∈
canarise,weshalloftenwrite insteadof . The properties(1)and(2)aboveimmediately
p
|·| |·|
translate asfollows:
(1’) x+y 6 max x , y and equality holds if x = y ,
| | | | | | | | 6 | |
(cid:0) (cid:1)
(2’) xy = x y .
| | | |·| |
Remark that (1’) implies that satisfies the triangular inequality, that is x + y 6 x + y
|· | | | | | | |
for all x,y Q . It is however much stronger: we say that the p-adic norm is ultrametric or
p
∈
nonArchimedean. We will see later that ultrametricity has strong consequenceson the topology
of Q (see for example Corollary 1.3 below) and strongly influences the calculus with p-adic
p
(univariateandmultivariate)functionsaswell(see 3.1.4). Thisisfarfrombeinganecdotic;on
§
the contrary, this will be the starting point of the theory of p-adic precision we will develop in
3.
§
The p-adic norm defines a natural distance d on Q as follows: we agree that the distance
p
between two p-adic numbers x and y is x y . Again this distance is ultrametric in the sense
p
| − |
that:
d(x,z) 6 max d(x,y),d(y,z) .
(cid:0) (cid:1)
Moreover the equality holds as soon as d(x,y) = d(y,z): all triangles in Q are isosceles! Ob-
p
6
serve also that d takes its values in a proper subset of R+ (namely 0 pn : n Z ) whose
{ }∪{ ∈ }
uniqueaccumulationpoint is 0. This property has surprising consequences;for example,closed
6
ballsofpositive radius are also open ballsandvice etversa. In particular Z is open (in Q ) and
p p
compact according to the topology defined by the distance. From now on, we endow Q with
p
this topology.
Clearly, a p-adic number lies in Z if and only if its p-adic valuation is nonnegative, that is
p
if and only if its p-adic norm is at most 1. In other words, Z appears as the closed unit ball
p
in Q . Viewed this way, it is remarkable that it is stable under addition (compare with R); it
p
is however a direct consequence of the ultrametricity. Similarly, by Proposition 1.1, a p-adic
integer is invertible in Z if and only if it has norm 1, meaning that the group of units of Z is
p p
then the unit sphere in Q . As for the maximal ideal of Z , it consists of elements of positive
p p
valuation and then appears as the open unit ball in Q (which is also the closed ball of radius
p
p−1).
1.1.4 Completeness
ThefollowingimportantpropositionshowsthatQ isnothingbutthecompletionofQaccording
p
to the p-adic distance. In that sense,Q arises in avery natural way... just asdoes R.
p
Proposition 1.2. The space Q equipped with its natural distance is complete (in the sense that
p
everyCauchysequenceconverges). MoreoverQ is densein Q .
p
Proof. We first prove that Qp is complete. Let (un)n>0 be a Qp-valued Cauchy sequence. It is
then bounded and rescaling the u ’s by a uniform scalar, we may assume that u 6 1 (i.e.
n n
| |
u Z ) for all n. For eachn,write :
n p
∈
∞
u = a pi
n n,i
Xi=0
with an,i 0,1,...,p 1 . Fix an integer i0 and set ε = p−i0. Since (un) is a Cauchy sequence,
∈ { − }
thereexistsarankN with thepropertythat u u 6 εforalln,m > N. Comingbackto the
n m
| − |
definitionofthep-adicnorm,wefindthatun umisdivisiblebypi0. Writingun = um+(un um)
− −
and computing the sum, we get an,i = am,i for all i 6 i0. In particular the sequence (an,i0)n>0
is ultimately constant. Let a 0,1,...,p 1 denote its limit. Now define ℓ = ∞ a pi Z
i0 ∈ { − } i=0 i ∈ p
andconsideragain ε> 0. Leti0 beanintegersuchthat p−i0 6 ε. ByconstructionP,thereexistsa
rankN forwhicha = a whenevern > N andi6 i . Forn > N,thedifferenceu ℓisthen
n,i i 0 n
−
divisible bypi0 and hencehas normat mostε. Hence(un) convergesto ℓ.
We nowprovethat Q is dense in Q . SinceQ = Z [1],it is enoughto prove that Z is dense
p p p p
in Z . Pick a Z and write a = a pi. For a nonnegative integer n, set b = n−1a pi.
p ∈ p i>0 i n i=0 i
Clearlybn is an integer andthe sequPence(bn)n>0 convergesto a. The density follows.P
Corollary 1.3. Let (un)n>0 be a sequence of p-adic numbers. The series n>0un converges in Qp
ifand onlyif its generaltermun convergesto 0. P
Proof. Set s = n−1u . Clearly u = s s for all n. If (s ) converges to a limit s Q ,
n i=0 i n n+1 − n n ∈ p
then un convergPes to s s = 0. We now assume that (un) goes to 0. We claim that (sn) is a
−
Cauchy sequence (and therefore converges). Indeed, let ε > 0 and pick an integer N for which
u 6ε for all i >N. Given two integersm andn with m > n > N, we have:
i
| |
m−1
s s = u 6 max u , u ,..., u
m n (cid:12) i(cid:12) n n+1 m−1
| − | (cid:12)(cid:12)Xi=n (cid:12)(cid:12) (cid:0)| | | | | |(cid:1)
(cid:12) (cid:12)
thanks to ultrametricity. Therefore(cid:12) s (cid:12)s 6 ε andwe are done.
m n
| − |
7
height
0
0 1 1
00 10 01 11 2
000 100 010 110 001 101 011 111 3
...010010001 Z
2
Figure1.3: Tree representationof Z
2
1.1.5 Tree representation
Geometrically,itisoftenconvenientandmeaningfultorepresentZ astheinfinitefullp-arytree.
p
In order to explain this representation,we needadefinition.
Definition 1.4. For h N anda Z ,we set:
p
∈ ∈
I = x Z s.t. x a (mod ph) .
h,a p
∈ ≡
(cid:8) (cid:9)
An intervalofZ is asubset ofZ ofthe form I for some hand a.
p p h,a
Ifadecomposesinbasepasa = a +a p+a p2+ +a ph−1+ ,theintervalI consists
0 1 2 h−1 h,a
··· ···
exactlyofthe p-adic integerswhose lastdigits area ...a a inthis order. On the otherhand,
h−1 1 0
from the analytic point of view, the condition x a (mod ph) is equivalent to x a 6 p−h.
≡ | − |
Thus the interval I is nothing but the closed ball of centre a and radius p−h. Even better, the
h,a
intervalsofZ areexactly the closedballsofZ .
p p
Clearly Ih,a = Ih,a′ if and only if a a′ (mod ph). In particular, given an interval I of Zp,
≡
there is exactly one integer h such that I = I . We will denote it by h(I) and call it the height
h,a
of I. We note that there exist exactly ph intervals of Z of height h since these intervals are
p
indexed by the classes modulo ph (or equivalently by the sequences of h digits between 0 and
p 1).
−
Fromthe topological point ofview, intervalsbehave like Z : they are atthe same time open
p
andcompact.
We now define the tree of Z , denoted by (Z ), as follows: its vertices are the intervals of
p p
T
Z and we put an edge I J whenever h(J) = h(I) + 1 and J I. A picture of (Z ) is
p 2
→ ⊂ T
representedonFigure1.3. Thelabelsindicatedontheverticesarethelasthdigitsofa. Coming
back to a generalp, we observe that the height of an interval I corresponds to the usual height
function in the tree (Z ). Moreover, given two intervals I and J, the inclusion J I holds if
p
T ⊂
andonlyif there exists a path from I to J.
ElementsofZ bijectivelycorrespondtoinfinitepathsof (Z )startingfromtherootthrough
p p
T
the following correspondence: an elementx Z is encodedby the path
p
∈
I I I I .
0,x 1,x 2,x h,x
→ → → ··· → → ···
8
Underthis encoding,an infinite path of (Z )starting from the root
p
T
I I I I
0 1 2 h
→ → → ··· → → ···
corresponds to a uniquely determined p-adic integer, which is the unique element lying in the
decreasingintersection I . ConcretelyeachnewI determinesanewdigitofx;thewhole
h∈N h h
collection of the Ih’s theTn defines x entirely. The distance on Zp can be visualized on (Zp) as
T
well: given x,y Z , we have x y = p−h where h is the height where the paths attached to
p
∈ | − |
x andy separate.
The above construction easily extendsto Q .
p
Definition 1.5. For h Z anda Q ,we set:
p
∈ ∈
I = x Q s.t. x a 6 p−h .
h,a p
∈ | − |
(cid:8) (cid:9)
Aboundedintervalof Q is asubset ofQ ofthe form I for some hand a.
p p h,a
Similarly to the case of Z , a bounded interval of Q of height h is a subset of Q consisting
p p p
ofp-adic numberswhosedigits atthepositions < harefixed(they haveto agreewith the digits
ofa at the same positions).
The graph (Q ) is defined as follows: its vertices are the intervals of Q while there is an
p p
T
edge I J if h(J) = h(I)+1 and J I. We draw the attention of the reader to the fact that
→ ⊂
(Q ) is a tree but it is not rooted: there does not exist a largest bounded interval in Q . To
p p
T
understandbetterthestructureof (Q ),letusdefine,foranyintegerv,thesubgraph (p−vZ )
p p
T T
of (Q )consistingofintervalswhicharecontainedinp−vZ . FromthefactthatQ istheunion
p p p
T
of all p−vZ , we derive that (Q ) = (p−vZ ). Moreover,for all v, (p−vZ ) is a rooted
p T p v>0T p T p
tree (with root p−vZp) which is isomoSrphic to (Zp) except that the height function is shifted
T
by v. The tree (p−v−1Z ) is thus obtained by juxtaposing p copies of (p−vZ ) and linking
p p
− T T
the roots ofthem to acommonparentp−vZ (which then becomesthe newroot).
p
1.2 Newton iteration over the p-adic numbers
Newtoniterationisawell-knowntoolinNumericalAnalysisforapproximatingazeroofa“nice”
functiondefined on arealinterval. Moreprecisely,givena differentiable function f :[a,b] R,
→
we define arecursivesequence(xi)i>0 by:
f(x )
i
x [a,b] ; x = x , i= 0,1,2,... (1.1)
0 ∈ i+1 i− f′(x )
i
Undersomeassumptions,onecanprovethatthesequence(x )convergestoazerooff,namely
i
x . Moreover the convergence is very rapid since, assuming that f is twice differentiable, we
∞
usually have an inequality of the shape x x 6 ρ2i for some ρ (0,1). In other words, the
∞ i
| − | ∈
number of correct digits roughly doubles at each iteration. The Newton recurrence (1.1) has
a nice geometrical interpretation as well: the value x is the x-coordinate of the intersection
i+1
pointof the x-axis with the tangentto the curvey = f(x)at the point x (seeFigure 1.4).
i
1.2.1 Hensel’s Lemma
It is quite remarkable that the above discussion extends almost verbatim when R is replaced
by Q . Actually, extending the notion of differentiability to p-adic functions is quite subtle and
p
probably the most difficult part. This will be achieved in 3.1.3 (for functions of class C1) and
§
3.2.3(forfunctionsofclassC2). Fornow,wepreferavoidingthesetechnicalitiesandrestricting
§
ourselves to the simpler (but still interesting) case of polynomials. For this particular case, the
9
x
∞
•x3 x2 x1 x0
Figure1.4: Newtoniteration over the reals
NewtoniterationisknownasHensel’sLemmaandalreadyappearsinHensel’sseminalpaper[36]
in which p-adic numbersare introduced.
Let f(X) = a + a X + + a Xn be a polynomial in the variable X with coefficients
0 1 n
···
in Q . Recall that the derivative of f can be defined in a purely algebraic way as f′(X) =
p
a +2a X + +na Xn−1.
1 2 n
···
Theorem 1.6 (Hensel’s Lemma). Let f Z [X] be a polynomial with coefficients in Z . We
p p
∈
supposethat we are given some a Z with the property that f(a) < f′(a)2. Then the sequence
p
∈ | | | |
(xi)i>0 definedby the recurrence:
f(x )
i
x = a ; x = x
0 i+1 i− f′(x )
i
is welldefinedandconvergesto x Z with f(x ) = 0. The rate ofconvergenceis givenby:
∞ p ∞
∈
2i
f(a)
x x 6 f′(a) | | .
| ∞− i| | |·(cid:18) f′(a)2(cid:19)
| |
Moreoverx is the unique rootoff in the openball ofcentre a andradius f′(a).
∞
| |
The proofof the above theorem is based on the nextlemma:
Lemma 1.7. Given f Z [X] andx,h Z , wehave:
p p
∈ ∈
(i) f(x+h) f(x) 6 h.
| − | | |
(ii) f(x+h) f(x) hf′(x) 6 h2.
| − − | | |
Proof. Foranynonnegativeintegeri,definef[i] = 1 f(i) wheref(i) standsforthei-thderivative
i!
off. Taylor’s formulathen reads:
f(x+h) f(x) = hf′(x)+h2f[2](x)+ +hnf[n](x). (1.2)
− ···
Moreover,adirectcomputation showsthat the coefficientsoff[i] areobtainedfromthat off by
multiplyingbybinomialcoefficients. Thereforef[i] hascoefficientsinZ . Hencef[i](x) Z ,i.e.
p p
∈
f[i](x) 6 1,foralli. Wededucethateachsummandoftherighthandside of(1.2)hasnormat
| |
most h. The first assertion followswhile the secondis provedsimilarly.
| |
Proofof Theorem1.6. Defineρ= |f(a)| . Wefirstprovebyinductiononithe followingconjunc-
|f′(a)|2
tion:
(H ) : f′(x ) = f′(a) and f(x ) 6 f′(a)2 ρ2i.
i i i
| | | | | | | | ·
10