Symmetric Diophantine approximation over function fields

184 Pages·2015·0.98 MB·English

Symmetric Diophantine

approximation over

function fields

Weidong Zhuang

Symmetric Diophantine
approximation over function
fields
Proefschrift
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden
op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker,
volgens besluit van het College voor Promoties
te verdedigen op donderdag 3 december 2015
klokke 10:00 uur
door
Weidong ZHUANG
geboren te Jiangsu, China
in 1983
Samenstelling van de promotiecommissie:
Promotor: Prof. dr. P. Stevenhagen
Copromotor: Dr. J.-H. Evertse
Overige leden:
Prof. dr. F. Beukers (Universiteit Utrecht)
Prof. dr. Y. Bugeaud (Universit´e Strasbourg)
Prof. dr. K. Gyo˝ry (University of Debrecen)
Dr. R. de Jong
Prof. dr. B. de Smit (secretaris)
Prof. dr. A.W. van der Vaart (voorzitter)
Mw. prof. dr. J. T.-Y. Wang (Academia Sinica Taiwan)
This work was funded by the NWO vrije competitie EW 2011 project
”Symmetric Diophantine Approximation” and was carried out at
Universiteit Leiden.
To Yana and Zhida

Contents
Introduction 1
1 Preliminaries 11
1.1 Discriminants and resultants . . . . . . . . . . . . . . . . . . 11
1.2 Valuations on function fields . . . . . . . . . . . . . . . . . . 13
1.3 Polynomials and heights . . . . . . . . . . . . . . . . . . . . 17
1.4 Galois theory of valuations . . . . . . . . . . . . . . . . . . . 18
1.5 Twisted heights . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Height estimates for solutions of S-unit equations 25
2.1 Height estimates . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 S-unit equations and heights . . . . . . . . . . . . . . . . . . 29
3 Geometry of numbers over function fields 47
3.1 Successive minima . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 A generalization . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Reduction theory for binary forms over k(t) 57
4.1 Discriminant and genus . . . . . . . . . . . . . . . . . . . . . 58
4.2 Preparations on polynomials . . . . . . . . . . . . . . . . . . 61
4.3 Reduced binary forms and successive minima . . . . . . . . . 65
i
ii Contents
5 Height estimates in terms of the discriminant 75
5.1 Consequences of the Riemann-Hurwitz formula . . . . . . . . 75
5.2 A few lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Completion of the Proof of the Main Theorem . . . . . . . . 101
6 Finiteness for the number of equivalence classes 105
6.1 GL(2,K)-equivalence classes . . . . . . . . . . . . . . . . . . 105
6.2 GL(2,O )-equivalence classes . . . . . . . . . . . . . . . . . 112
S
7 Lower bounds for resultants 115
7.1 Monic binary forms . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Results for binary cubic forms . . . . . . . . . . . . . . . . . 120
7.3 Binary forms of arbitrary degree . . . . . . . . . . . . . . . . 126
7.4 A result on Thue-Mahler equations . . . . . . . . . . . . . . 131
7.5 Lower bounds for resultants in terms of heights . . . . . . . 134
8 Distances between algebraic functions 139
8.1 Root separation of polynomials . . . . . . . . . . . . . . . . 139
8.2 Two lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.3 A symmetric improvement of the Liouville-type inequality . 149
Bibliography 155
Abstract 159
Samenvatting 161
Acknowledgements 167
Curriculum Vitae 169
Index 171
Introduction
Roth’stheoremgivesanoptimalsolutiontotheproblemhowwellagivenal-
gebraicnumbercanbeapproximatedbyotheralgebraicnumbers. Anatural
question is to ask how well two varying algebraic numbers can approximate
each other. There is only one non-trivial result, proved by Evertse, but
this is far from optimal. Its proof is based on a weak version of the abc-
conjecture, which is a consequence of a generalization of Roth’s Theorem,
hence it is non-effective.
Let k be an algebraically closed field of characteristic 0. Over algebraic
function fields of transcendence degree 1 over k there is a proved analogue
of the abc-conjecture, i.e., the Mason-Stothers Theorem. This suggests
that it should be possible to develop much stronger symmetric Diophantine
approximation results over function fields. My research focuses mainly on
this interesting problem.
To tackle this problem, one considers two cases: either the two algebraic
functionsthatapproximateeachotherareconjugateoverthefieldofrational
functions k(t) or not.
The first case is strongly connected to the following problem: over the
Z
integers, two binary forms (i.e., homogeneous polynomials) F,G ∈ [X,Y]
are called equivalent if G(X,Y) = F(aX +bY,cX +dY) for some matrix
(cid:0)a b(cid:1) ∈ GL(2,Z). Twoequivalentbinaryformshavethesamediscriminant.
c d
A binary form F is called reduced if its height H(F) (maximum of the
absolutevaluesofitscoefficients)isminimalamongtheheightsofthebinary
1
2 Introduction
forms in its equivalence class.
(cid:62)
Conjecture. The height H(F) of a reduced binary form F of degree n 4
and non-zero discriminant D has an upper bound of the form c1(n)|D|c2(n),
where c (n), c (n) are numbers depending only on n.
1 2
An analogous estimate for n = 2 and n = 3 follows from work of La-
grange, Gauss and Hermite. However, the general case is still open. There
is only the following much weaker effective result from [11]:
Z
Theorem (Evertse, Gyo˝ry). Let F(X,Y) ∈ [X,Y] be a reduced binary
(cid:62)
form of degree n 2 and discriminant D(F) (cid:54)= 0. Then
H(F) (cid:54) exp((c n)c2n4|D|8n3),
1
where c ,c are effectively computable, absolute constants.
1 2
More generally, we may consider the ring of integers of an algebraic
Z
number field and even the ring of S-integers instead of . A weak version
of Evertse [9] implies the following:
Z
Theorem (Evertse). Let F ∈ [X,Y] be a reduced binary form of degree
Q
n > 1 with splitting field L over and non-zero discriminant. Then
H(F) (cid:54) Cineff(n,L)|D(F)|n2−11.
The constant here depends on n,L and is ineffective in the sense that it
is not effectively computable from the method of proof. We call this result
a ’semi-effective’ upper bound since it is effective in terms of D(F), but
ineffective in terms of n and L.
We proved an analogue of the above conjecture over k[t]. Our main tools
areananalogueofthegeometryofnumbersoverfunctionfields(seeThunder
[24]) and Mason’s theorem which is an analogue of the abc-conjecture over
function fields.