Polynomials in finite geometry

262 Pages·2016·1.755 MB·English

by  Sziklai, Peter

POLYNOMIALS IN FINITE GEOMETRY
Peter Sziklai
20 February, 2008
ii
Foreword iii
-1 Special foreword
Before the real Foreword, let me say some words about this book. It is ready in
the sense that this is the topic that I wanted to write about and this is how I
wanted to write about it. Naturally, this book is not finished in the sense that (i)
it is about an active field of mathematics, which changes rapidly; (ii) if you have
worked on a book for years then it is hard to stop it: every day you can have a
new idea how to slightly improve it; (iii) no editor/referees have seen it yet, hence
there may (must!) be typos, inaccuracies, missing citations, line overflow errors...
left. It would be nice to increase the number of figures as well.
However, I think that in the current state of this volume it is possible to decide
aboutitsvaluesandshortcomings.AlthoughIunderstandthatitcanbeevaluated
by this current version, in the spirit of (i) and (ii) above I will keep an updated
version on the secret, unlinked webpage
http://www.cs.elte.hu/˜sziklai/poly.html , (where I will enlist all the
changes performed, too), just for pleasure. When a decision of publication is
reached, the editors of the publisher can decide between this and a possibly up-
dated version.
0 Foreword
Amostefficientwayofinvestigatingcombinatoriallydefinedpointsetsofprojective
spacesoverfinitefieldsisassociatingpolynomialstothem.Thistechniquewasfirst
used by Jamison and Bruen, then, followed by several people, became a standard
method; nowadays, the contours of a growing theory can be seen already.
Thepolynomialsweuseshouldreflectthecombinatorialpropertiesofthepointset,
thenwehavetobeequippedwithenoughmeanstohandleourpolynomialsandget
an algebraic description about them; finally, we have to translate the information
gained back to the original, geometric language.
The first investigations in this field examined the coefficients of the polynomials,
and this idea proved to be very efficient. Then the derivatives of the polynomials
came into the play and solving (differential) equations over finite fields; a third
branchofresultsconsideredthepolynomialsasalgebraiccurves.Theideaofasso-
ciating algebraic curves to pointsets goes back to Segre, recently a bunch of new
applicationshaveshownthestrengthofthismethod.Finally,duetoG´acs’srecent
results, dimension arguments on polynomial spaces have become fruitful.
This book is an attempt to collect and classify the most interesting ways of ap-
plying polynomials in finite geometry.
We focus on combinatorially defined (point)sets of projective geometries. They
aredefinedbytheirintersectionnumberswithlines(orothersubspaces)typically,
like arcs, blocking sets, nuclei, caps, ovoids, etc. Here we do not want to give a
iv
complete account on them, however some of the best results will be presented.
In order to show a wider scope of different applications, some other fields are
gently touched like group theory, graph theory, etc.
This book is divided into chapters. The first one (Background) contains selected
toolsforafinitegeometer,fromthebasicfactstosometheoryofpolynomialsover
finitefields;proofsareonlyprovidedwhentheyareshortorinterestingforus.We
provide slightly more information than the essential background for the second
chapter. After the basic facts (Section 5) the most useful representations of affine
andprojectivespacesarepresented(6),thenweintroduceourmaintool,theR´edei
polynomialassociatedtopointsets(7,8).ThecoefficientsofR´edeipolynomialsare
elementarysymmetricpolynomialsthemselves,whatweneedtoknowaboutthem
and other invariants of subsets of fields or spaces is collected in Section 9. The
multivariate polynomials associated to pointsets can be considered as algebraic
varieties, so we can use some basic parts of algebraic geometry (10). Finally, in
Section 11 some background needed for stability results is presented.
The second (and main) chapter contains several results of finite Galois geometry,
where polynomials play a main role. We start with general results on intersection
numbers of planar pointsets (12,13). Then we turn to special cases as arcs, max-
imal arcs, unitals, semiovals, untouchable sets. (14-17). The next highlight is the
topic of directions (18,19) and blocking sets (20-22). Section 23 shows the new
resultant method for stability results. Affine blocking sets and nuclei are consid-
ered in sections 24,25. Section 26 introduces an interesting mixed representation.
Stability theorems for flocks, some basic facts for spreads and a nice result on
ovoids is proved in Sections 27,28. Finally some non-geometrical applications are
collected in Section 29.
The last chapter contains hints and solutions for the exercises.
Contents
-1 Special foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
0 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Definitions, basic notation . . . . . . . . . . . . . . . . . . . . . . . 3
1 Background 5
5 Finite fields and polynomials . . . . . . . . . . . . . . . . . . . . . 5
5.1 Some basic facts . . . . . . . . . . . . . . . . . . . . . . . . 5
5.2 Self-dual normal bases . . . . . . . . . . . . . . . . . . . . . 6
5.3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5.4 Differentiating polynomials . . . . . . . . . . . . . . . . . . 8
5.5 Polynomials vanishing at many points, Alon’s Combinato-
rial Nullstellensatz and the Ball-Serra refinement . . . . . . 9
5.6 Additive, linearized polynomials . . . . . . . . . . . . . . . 16
5.7 The random-like behaviour . . . . . . . . . . . . . . . . . . 17
5.8 Lacunary polynomials . . . . . . . . . . . . . . . . . . . . . 19
6 Representations of affine and projective geometries . . . . . . . . . 22
6.1 Subspaces, subgeometries . . . . . . . . . . . . . . . . . . . 24
6.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . 25
7 The R´edei polynomial and its derivatives . . . . . . . . . . . . . . 26
7.1 The R´edei polynomial . . . . . . . . . . . . . . . . . . . . . 26
7.2 Differentiation in general . . . . . . . . . . . . . . . . . . . 30
7.3 Hasse derivatives of the R´edei polynomial . . . . . . . . . . 32
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Univariate representations . . . . . . . . . . . . . . . . . . . . . . . 35
v
vi Contents
8.1 The affine polynomial and its derivatives. . . . . . . . . . . 36
8.2 The projective polynomial . . . . . . . . . . . . . . . . . . . 37
8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9 Symmetric polynomials and invariants of subsets . . . . . . . . . . 39
9.1 The Newton formulae . . . . . . . . . . . . . . . . . . . . . 39
9.2 The invariants u ,v and w . . . . . . . . . . . . . . . . . 43
f f f
9.3 Arbitrary subsets . . . . . . . . . . . . . . . . . . . . . . . . 47
9.4 The generalized formulae . . . . . . . . . . . . . . . . . . . 51
9.5 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 Elements of algebraic geometry . . . . . . . . . . . . . . . . . . . . 61
10.1 Conditions implying linear (or low-degree) components. . . 65
11 Finding the missing factors, removing the surplus factors. . . . . . 68
2 Polynomials in geometry 71
12 Lines meeting a pointset . . . . . . . . . . . . . . . . . . . . . . . . 71
12.1 Prescribing the intersection numbers . . . . . . . . . . . . . 71
12.2 A general result. . . . . . . . . . . . . . . . . . . . . . . . . 72
12.3 Another immediate corollary . . . . . . . . . . . . . . . . . 73
13 Sets with constant intersection numbers mod p . . . . . . . . . . . 74
13.1 Sets with intersection numbers 0 mod r . . . . . . . . . . . 76
13.2 Small and large super-Vandermonde sets . . . . . . . . . . . 80
14 Arcs, Segre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
15 Maximal arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
15.1 Existence: hyperovals . . . . . . . . . . . . . . . . . . . . . 89
15.2 Non-existence . . . . . . . . . . . . . . . . . . . . . . . . . . 93
15.3 Embeddability . . . . . . . . . . . . . . . . . . . . . . . . . 96
16 Unitals, semiovals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
17 Sets without tangents: where Segre meets R´edei . . . . . . . . . . . 102
17.1 The proof using Segre’s method . . . . . . . . . . . . . . . . 103
17.2 Untouchable sets in Galois planes of even order . . . . . . . 109
18 Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
18.1 When D is (contained in) a subgroup of GF(q)∗ . . . . . . . 112
√
18.2 When D is (contained in) a subgroup of q+11⊂GF(q2)∗ . 115
18.3 Sets determining ≤ q+1 directions: affine linear pointsets . 115
2
18.4 Small blocking sets of R´edei type are linear . . . . . . . . . 116
Contents vii
19 Over prime fields: the G´acs method. . . . . . . . . . . . . . . . . . 122
19.1 Permutation polynomials . . . . . . . . . . . . . . . . . . . 124
19.2 Directions in the plane . . . . . . . . . . . . . . . . . . . . . 125
19.3 Linear combinations of three permutation polynomials . . . 131
20 Projective linear pointsets . . . . . . . . . . . . . . . . . . . . . . . 134
21 Blocking sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
21.1 One curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
21.2 Three new curves . . . . . . . . . . . . . . . . . . . . . . . . 144
21.3 Three old curves . . . . . . . . . . . . . . . . . . . . . . . . 147
21.4 Small blocking sets . . . . . . . . . . . . . . . . . . . . . . . 149
22 Multiple blocking sets . . . . . . . . . . . . . . . . . . . . . . . . . 154
22.1 A t (mod p) result . . . . . . . . . . . . . . . . . . . . . . . 155
23 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
23.1 Stability theorems for small blocking sets and the Sz˝onyi-
Weiner method . . . . . . . . . . . . . . . . . . . . . . . . . 164
23.2 Stability theorems for blocking sets: the prime case . . . . . 167
23.3 Stability theorems for blocking sets: the non-prime case . . 170
23.4 Stability theorem for sets of even type . . . . . . . . . . . . 176
23.5 Number of lines meeting a (q+2)-set . . . . . . . . . . . . 179
24 Jamison, Brouwer-Schrijver, affine blocking sets . . . . . . . . . . . 180
24.1 Applications of the punctured Nullstellensatz . . . . . . . . 184
24.2 R´edei polynomials in three indeterminates . . . . . . . . . . 189
24.3 R´edei polynomials in many indeterminates . . . . . . . . . 191
24.4 Affine blocking sets. . . . . . . . . . . . . . . . . . . . . . . 195
25 Nuclei and affine blocking sets. . . . . . . . . . . . . . . . . . . . . 198
25.1 Lower nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . 201
25.2 Internal nuclei . . . . . . . . . . . . . . . . . . . . . . . . . 203
25.3 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 203
26 Mixed representations . . . . . . . . . . . . . . . . . . . . . . . . . 205
27 Flocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
27.1 Partial flocks of the quadratic cone in PG(3,q) . . . . . . . 208
27.2 Partial flocks of cones of higher degree . . . . . . . . . . . . 211
28 Spreads and ovoids . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
28.1 Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
28.2 Ovoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
viii Contents
29 Non-geometrical applications . . . . . . . . . . . . . . . . . . . . . 219
29.1 Normal factorizations of Abelian groups . . . . . . . . . . . 219
29.2 The Paley graph . . . . . . . . . . . . . . . . . . . . . . . . 220
29.3 Representing systems . . . . . . . . . . . . . . . . . . . . . 221
29.4 Wielandt’s visibility theorem . . . . . . . . . . . . . . . . . 222
29.5 Burnside’s theorem . . . . . . . . . . . . . . . . . . . . . . . 222
29.6 Jaeger’s conjecture . . . . . . . . . . . . . . . . . . . . . . . 223
29.7 Graphs containing p-regular subgraphs . . . . . . . . . . . . 225
29.8 Blokhuis’ proof of Bollob´as’ theorem . . . . . . . . . . . . . 225
29.9 Alon-Fu¨redi . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
29.10 Chevalley-Warning . . . . . . . . . . . . . . . . . . . . . . . 226
29.11 Cauchy-Davenport . . . . . . . . . . . . . . . . . . . . . . . 227
29.12 Another application . . . . . . . . . . . . . . . . . . . . . . 228
3 Hints and short solutions to the exercises 231
1 Of Chapter I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
2 Of Chapter II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4 Glossary of concepts 241
5 Notation 243
Index 244
References 245
1. Introduction 1
1 Introduction
This one is not intended to be an introductory textbook about finite geometries,
finite fields nor polynomials. There are very good books of these kinds available,
e.g. Ball-Weiner [23] for a smooth and fascinating introduction to the concepts of
finite geometries, the three volumes of Hirschfeld and Hirshfeld-Thas [77, 78, 79]
ashandbooksandLidl-Niederreiter[90]forfinitefields.Still,theinterestedreader,
even with a little background, may find all the definitions and basic information
here (the Glossary of concepts at the end of the volume can help) to enjoy this
interdisciplinary field in the overlap of geometry, combinatorics and algebra.
We would like to use a common terminology. Parts and reformulations of results
of some interest for non-geometers will be emphasized.
Bruen and Fisher called the polynomial technique as “the Jamison method” and
summarized it in performing three steps: (1) Rephrase the theorem to be proved
as a relationship involving sets of points in a(n affine) space. (2) Formulate the
theoremintermsofpolynomialsoverafinitefield.(3)Calculate.(Obviously,step
3 carries most ofthe difficulties in general.) In some sense it is still “the method”,
we will show several ways how to perform steps 1-3.
We have to mention the book of L´aszl´o R´edei [103] from 1970, which inspired a
new series of results on blocking sets and directions in the nineties.
Thereareafewsurveypapersonpolynomialmethods,byBlokhuis[35,36],Sz˝onyi
[122], Ball [7], I included here some of their material. The form of a book allows
to show the algebraic background in more details; also to insert the most recent
results.
The typical theories in this book have the following character. Define a class of
(point)sets (of a geometry) in a combinatorial way (which, in general, means re-
strictions on the intersections with subspaces); examine its numerical parameters
(usuallythespectrumofsizesintheclass);findtheminimal/maximalvaluesofthe
spectrum; characterize the extremal entities of the class; show that the extremal
(or other interesting) ones are “stable” in the sense that there are no entities of
the class being “close” to the extremal ones.
There are some fundamental concepts and ideas that we feel worth to put into
light all along this treatise:
• an algebraic curve (or envelope) containing all the “interesting” lines of a
pointset;
• the Lemma of tangents with its generalizations and analogues;
• resultants as sophisticated tools improving the simple use of B´ezout’s theo-
rem;
• examination of (lacunary) coefficients of polynomials;
2 Introduction
• considering (sub)spaces generated by polynomials.
Apology. There are some theorems being proved more than once in this book,
these multiple proofs were included in order to show how the different ideas and
techniques can lead to the same result; sometimes they even serve as touchstones
of a new method.
Also, this book contains a few proofs being longer than the usual ones. They
were included to show some elaborate series of ideas that lead to practical use of
polynomial techniques.
There are exercises, like a hundred and twenty. One type of them is for the neces-
sary technicalities where no further idea is needed (so boring but useful), another
typeiswhereanargumentanalogoustotheprecedingonescansolvethequestion
(so good to practice), and a third type which needs minor new ideas (so the in-
teresting ones). We provide short solutions for a great part of them at the end of
the volume.
There are side comments on the margins. They might help reading, the edi-
tors/referees are kindly asked to decide whether they should be (i) left as they
Please!
are, or (ii) included into the normal text, or (iii) the number of them can be
increased (I would be happy to add more).
2 Acknowledgements
Most of this book is about the work of Simeon Ball, Aart Blokhuis, Andr´as G´acs,
Tam´as Sz˝onyi and Zsuzsa Weiner. They, together with the author, contributed in
roughlyonehalfofthereferences;alsotheirresults(andsometimeseventheirtexts
withminormodifications:thankstothemforlettingmedoso)formanimportant
partofthisvolume.Notleast,Ialwaysenjoyedtheirwarm,joyful,inspiratingand
supporting company in various situations in the last some years. I hope that the
selection and the choice of the topics covered in this book makes them happy as
well, and that being the “topic” of a book is at least as funny as writing it. Most
of them have seen a preliminary version of it, I am grateful for all the suggestions
they made.
AboveallIamdeeplyindebtedtoTam´asSz˝onyi,fromwhommostofmyknowledge
and most of my enthusiasm for finite geometries I have learned.
It was G´abor Korchm´aros who suggested (and not only to me) some seven years
ago to write a book like this. Probably any of us, or maybe someone else, could
have done it. Of course this one is about the way how I can see the topic.
Last but not least I am grateful to all the colleagues and friends who helped me
in any sense in the last some years: researchers of Ghent (and among them my
multiple coauthor Leo Storme), Potenza, Naples, Barcelona, Eindhoven and, of
course, Budapest.