Some new results on semiarcs in finite projective planes and on inverse-closed subsets in fields

Transcription

1 Universitá degli Studi della Basilicata Some new results on semiarcs in finite projective planes and on inverse-closed subsets in fields Dottorato di Ricerca in Matematica János Bolyai MAT 03 Coordinatore Prof. Gábor Korchmáros Dottorando Dott. Bence Csajbók Tutor Prof. Gábor Korchmáros A.A. 2012/2013 Ciclo XXVI

2 Acknowledgements First of all, I am very grateful to my supervisor Gábor Korchmáros for the numerous things I have learnt from him and for the topic of the second part of my thesis. His great ideas on what articles I should read and how I should write, helped me a lot. I am also very grateful to György Kiss, who was the supervisor of my master s thesis. His support and our collaboration remained constant during my doctorate years. Studying semiarcs was his idea, for which I am also very grateful. I would also like to thank Tamás Szőnyi for all his help and suggestions. His ideas on what the right bound might be and what articles I should read, always turned out to be right. From Potenza, I would also like to thank my colleague and friend Francesco Pavese for delicious pastas, many mathematical discussions, his help in administrative matters and for the great company during the years I spent in Potenza. I am also very thankful to Alessandro Siciliano, who always had time to answer my questions. I am also grateful to both of them for their help with computer programming. From Budapest, I would also like to thank the members of the MTA-ELTE Geometric and Algebraic Combinatorics Research Group for the inspiring and peaceful atmosphere, which I had as a group member during the last month of the preparation of this thesis. Special thanks to my co-author Tamás Héger for the collaboration, in particular for sharing his knowledge on the Szőnyi-Weiner Lemma. I would also like to express my gratitude to Sandro Mattarei for inviting me to Trento and for many mathematical discussions. My participation to conferences was funded by the Department of Mathematics at University of Basilicata and by the Hungarian National Foundation for Scientific Research, Grant No. K I am thankful for the support of M.I.U.R. as well, which made my PhD studies possible in Italy. Finally, I want to thank the support of my family and Brigi.

3 Abstract This thesis consists of two main parts. A semiarc in a finite projective plane is a point set having the same number of tangents through each of its points. In the first part we study semiarcs. We give constructions and characterization results. Relations with other objects in finite geometry, such as blocking sets, point sets determining few directions, ovals, linear spaces and codewords, are also discussed. We call a subset of a field inverse-closed, if it is closed with respect to inverting non-zero elements. In the second part of our thesis we investigate the maximal possible size of an inverse-closed subset contained in an additive subgroup of a finite field. We also consider several generalizations of this problem. Examples showing sharpness of our bound are provided using ideas from finite geometry.

4 Contents Contents iii 1 Introduction 1 2 Preliminary definitions and results Finite fields and polynomials Projective and affine planes Special point sets in finite projective and affine planes Projective and affine spaces over a finite field Associated polynomials The Hasse Bound and the Lang Weil estimate Collineations Results on minimal blocking sets Rédei type blocking sets and the direction problem Affine blocking sets, covers Perspective point sets Results from additive combinatorics Difference sets Semiarcs in finite projective planes Semiovals Definition and basic properties of semiarcs Large collinear subsets in semiarcs The Desarguesian case Semiarcs with two long secants Semiarcs containing a V t -configuration in PG(2, q) Semiarcs with index three iii

5 CONTENTS Semiarcs in the sides of a triangle Semiarcs contained in the union of three concurrent lines The spectrum of the sizes of semiarcs with index three Strong semiovals and semiovals of size 3(q p m ) Large collinear subsets in small semiarcs Semiarcs and the direction problem A lemma on tangents Small semiarcs and small blocking sets Connections with (n + 1, 1)-designs Small semiovals Codewords and the symmetric difference of lines Small blocking semiovals and semi-resolving sets Inverse-closed subsets in fields Introduction A bound on the size of B A Examples and their geometry Examples in PG(3, q) Characterization of the examples Higher dimensions Infinite fields References 101 iv

6 Chapter 1 Introduction My work as a PhD student at the University of Basilicata can be divided into two main parts. Let us first introduce semiovals and semiarcs. A semioval in a finite projective plane Π is a non-empty point set S with the property that through every point P of S, there passes a unique line of Π meeting S exactly in the point P. This line is called the tangent at P to S. The definition of semiovals is due to Buekenhout. The first results on semiovals were obtained by Thas, who showed that ovals and unitals are the smallest and the largest semiovals respectively. Both objects have been widely studied by finite geometers, but their complete classification is still open, even in Desarguesian planes. What can we say about semiovals, if we cannot even classify those of extremal size? In Section 3.1 we present some of other authors results to illustrate what kind of problems have been investigated in this area. After this, we introduce the notion of t-semiarcs. These objects are natural generalizations of semiovals. A t-semiarc in Π is a non-empty point set S t having t-tangents at each of its points, where t is a positive integer. Chapter 3 has been devoted to the study of semiarcs. With our techniques developed for semiarcs, we were able to obtain new results and to improve previous ones, also on semiovals. Besides combinatorial arguments, in Desarguesian planes we also use algebraic methods. Our techniques have limited us to consider only small semiarcs of PG(2, q). In general, we mostly deal with semiarcs of size less than 3q. In this sense, the situation is similar to the case of blocking sets. Also, small blocking sets can be constructed from small semiarcs, and vice versa, as we will see in Section 3.6. In our proofs we will also use results on small minimal blocking sets. Let us outline the main topics of Chapter 3. In Section 3.2 we generalize Thas bound for semiarcs, examine the case of equality 1

7 and present some elementary facts and examples. In Section 3.3 we give the semiarc analogous of a combinatorial result by Dover on the maximal size of a collinear subset contained in a semioval. We also give a stronger bound in PG(2, q), which is sharp only for Rédei type blocking sets of size 3(q + 1)/2. Our proof relies on a result by Szőnyi and Weiner, which has been conjectured by Metsch. Combining this, with wellknown results on blocking sets due to Gács, Lovász, Schrijver and Szőnyi, we get a characterization of the projective triangle. We also have to mention Blokhuis result which states that the smallest minimal non-trivial blocking sets in PG(2, p), p prime, are (p 1)/2-semiarcs. We call the symmetric difference of two lines, with t further points removed from both lines, a V t -configuration. In Section 3.4 we give conditions ensuring a t-semiarc to contain a V t -configuration and we give the complete description of such t-semiarcs in PG(2, q). This result will be used several times in the next sections. It can be viewed as the semiarc analogous of the description of blocking sets with two Rédei lines. Our proof relies on the classification of perspective point sets in PG(2, q), which is a result due to Korchmáros and Mazzocca, related to Dickson s classification of the subgroups of AGL(1, q). Semiovals in PG(2, q) contained in the union of three lines were investigated by Kiss and Ruff. In the first part of Section 3.5 we use combinatorial arguments to extend some of their results to semiarcs. Our examples in PG(2, q) arise from cyclic difference sets, skew Hadamard difference sets and partial difference sets. At the end of this section we use theorems from additive combinatorics to prove spectrum results on the sizes of semiarcs contained in the union of three concurrent lines. More precisely, we use results on t-representable sums, such as Pollard s Theorem and its recent generalization by Grynkiewicz. As far as we know, this is the first application of these results in finite geometry. Strong semiovals were defined by Blokhuis, Kiss, Kovács, Malnič, Marušič and Ruff, as special semiovals contained in the union of three concurrent lines. As a by-product of our results, we managed to exclude the existence of strong semiovals in PG(2, p h ), p > 7 prime, h odd, and to prove that strong semiovals in PG(2, q) have size 3(q q), when q is an odd square. One can obtain a t-semiarc from the symmetric difference of a Baer subplane B and a line l, by removing t further points from B \ l and from l \ B. In Section 3.6 we prove that small semiarcs having a large collinear subset K necessarily arise in this way, or contain a V t -configuration. This question is strongly related to point sets determining few directions and to small minimal blocking sets. The main part of our proof is a lemma, which states that the tangents at the points of K are contained in 2

8 t pencils with carriers not in K. We hope that this lemma can be applied to attack other problems as well. Our proof is based on an application of the Rédei-polynomials and the Szőnyi Weiner Lemma on polynomials. We also use some deep results from the area of blocking sets. (n + 1, 1)-designs are linear spaces with constant point degree n + 1. In Section 3.7 we briefly discuss connections between t-semiarcs and (n + 1, 1)-designs. We also reformalize some of our theorems to obtain results on (n + 1, 1)-designs which can be embedded into PG(2, q), q > n. In Section 3.8 we present a characterization of ovals in PG(2, q), q even, as semiovals of size at most q + 3 q 11. The analogous problem in planes of odd order is related to the symmetric difference of lines, which has been investigated recently by Balister, Bollobás, Füredi and Thompson. Relations between blocking semiovals and semi-resolving sets in PG(2, q) are also discussed. Using a result by Héger and Takáts, we present a new lower bound on the size of a blocking semioval in PG(2, q), q = p h, where p is a prime greater than 5. The results of this section also rely on the Szőnyi Weiner Lemma, which turned out to be a powerful tool in the study of the behavior of tangents. The second part of our work is presented in Chapter 4. This is a question from algebra, which has some interesting geometric aspects. We call a subset of a field inverse-closed, if it is closed with respect to inverting non-zero elements. Mattarei investigated those additive subgroups of a field which are inverse-closed. Independently Goldstein, Guralnick, Small and Zelmanov answered the more general question with division rings instead of fields. Similar questions were also investigated by Kroll, related to chain geometries. In Section 4.2 we prove that an additive subgroup of a finite field with a large inverse-closed subset is necessarily inverse-closed. This is obtained as the special case A = B and q = p of the following more general result. Let A and B be two GF(q)- subspaces of a finite field of characteristic p, of the same size. If the set of inverses of the non-zero elements of A shares at least 2 B /q 1 elements with B, then they are both one-dimensional subspaces over the same subfield. If q = 2, then the above result holds under a weaker condition. This research was also motivated by an article by Korchmáros, Lanzone and Sonnino, related to 2-level secret-sharing schemes. Our results can also be applied to construct mixed partitions of PG(3, q), as Lavrauw and Zanella did. In Section 4.3 we investigate the case of equality in our bound, when A and B are 3-dimensional GF(q)-subspaces of GF(q 4 ). It turns out that these examples can be 3

9 described using an elliptic quadric constructed by Ebert, as an orbit of a subgroup of a Singer group of PG(3, q). Using the Hasse Bound on the number of rational points of an algebraic curve, we give another characterization of these examples. We also discuss briefly an improvement by Mattarei, depending on the Lang Weil estimate, for higher dimensional subspaces. In the last section we give a possible generalization to infinite fields. The elementary proof depends on Hua s identity. This thesis is based on two published articles, one extended abstract and two submitted manuscripts. In the last few years we have improved some of our published results, while some of our other results have not been published yet. In this thesis we present the currently best results. Sections 3.3 and 3.4 are based on [28, 30]. Sections 3.2 and 3.5 is based on [31], which is joint work with György Kiss. Section 3.6 relies entirely on [32], which is joint work with Tamás Héger and György Kiss. Finally, Chapter 4 presents [29]. In Chapter 2 we collected definitions, other authors results and some folklore results. 4

10 Chapter 2 Preliminary definitions and results 2.1 Finite fields and polynomials We denote by GF(q) the finite field of order q. For arbitrary fields we will use the notation F and K. The algebraic closure of F is denoted by F. For a subset S of a field F let S be the set of all non-zero elements of S. By S 1 we denote the inverse set of S, that is, S 1 = {s 1 : s S }. If S 1 = S, S is called inverse-closed. The multiplicative group of GF(q) is a cyclic group, which we denote by GF(q). The generators of GF(q) are called the primitive elements of GF(q). The additive group of GF(q) is an elementary abelian p-group, where p is the characteristic of GF(q), i.e. q = p h for some positive integer h. We denote this group by GF(q) +. For any field F, by F[X 1, X 2,..., X n ] we denote the polynomial ring in n variables over F. Definition Let F be an arbitrary field of characteristic p > 0 and let q be a p-power. We call a polynomial f(x) F[x] q-polynomial if it is of the form: f(x) = m a i x qi. i=0 If q is clear from the context, as in our case, we call them linearized polynomials. If F is a finite field, there is a vast literature on linearized polynomials over F, see for example the book of Lidl and Niederreiter [72, Chapter 4]. In Chapter 3 our main tool will be the following theorem. For a proof, see for example the book by Goss [48]. 5

11 Theorem ( Fundamental Theorem of Additive Polynomials [48, Corollary 1.2.2]) If F is a field of characteristic p > 0, f(x) F[x] and it splits into deg f distinct linear factors, then f(x) is a q-polynomial if and only if its roots form a GF(q)-subspace of F. We give here the proof of two easy and well-known results. Proposition (Folklore). Let F be a field of p > 0 characteristic and S a subset of F of size p d for some positive integer d. If s S (x s) = xpd tx for some t F, then S is a (one-dimensional) GF(p d )-subspace. Proof. Suppose that s S (x s) = xpd tx for some t F and take a non-zero element c S. We prove that c 1 S = {c 1 s: s S} is the subfield GF(p d ). To show this, it is enough to see a pd element b S satisfying a = c 1 b. Therefore: = a for every a c 1 S. Let a c 1 S, so there is an a pd = c pd b pd = (tc) 1 tb = c 1 b = a. We conclude S = c GF(p d ) = {cf : f GF(p d )}. For a polynomial f(x) = n i=0 a ix i with a 0 a n 0 the reciprocal polynomial of f(x) is x n f(1/x) = n i=0 a n ix i. Clearly, if Z f is the set of roots of f(x), then Z 1 f is the set of roots of its reciprocal polynomial with the corresponding multiplicities. We use this result in the following way. Proposition (Folklore). If S is a non-empty finite subset of non-zero elements in the field F and f(x) = s S (x s), with constant term a 0 0 and deg f = S = n, then s S 1(x s) = a 1 0 xn f(1/x). Proof. x n f(1/x) is the reciprocal polynomial of f(x) and hence its set of roots is S 1. The polynomial p(x) := s S 1(x s) a 1 0 xn f(1/x) has at least n distinct roots while its degree is less than n. If F is an infinite field, then p(x) = 0 follows immediately. In the case when F is the finite field of order q we have p(x) 0 (mod X q X) in the polynomial ring GF(q)[x]. Since deg p < n < q, this also implies p(x) = Projective and affine planes An incidence structure is a triple (P, L, I), where P is a set of points, L a set of lines and I P L an incidence relation between them. The elements of I are also called 6

12 flags. We will only consider finite incidence structures, that is, when A and B are finite sets. Definition An affine plane is an incidence structure satisfying the following three axioms. (A1) Every two points are incident with a unique line. (A2) For every point P and line l there exists a unique line through P not intersecting l. (A3) There exist three distinct points, no three of which are collinear. Definition A projective plane is an incidence structure satisfying the following three axioms. (P1) Every two points are incident with a unique line. (P2) Every two lines are incident with a unique point. (P3) There exist four distinct points, no three of which are collinear. Note that the axioms (P1)-(P3) are self-dual, hence if a statement can be deduced from these axioms, then the dual statement is also true (that is, when we interchange the role of points and lines in the statement). Linear spaces are the generalizations of affine and projective planes. Definition A linear space (P, L, I) is an incidence structure satisfying the following three axioms. (L1) Every two points are incident with a unique line. (L2) Every line contains at least two points. (L3) There are at least two lines. The degree of a point P P is the number of lines incident with P, and, dually, the degree of a line l L is the number of points incident with l. If n + 1 is the maximum point degree, then n is called the order of (P, L, I). Two lines, l 1 and l 2, are said to be parallel if l 1 l 2 and they do not intersect each other. An embedding of (P, L, I) into a linear space (P, L, I ) is a function π mapping P into P and L into L which is one-to-one in both points and lines, and such that for each P P and l L we have P Il if and only if π(p )I π(l). 7

13 Definition An (n + 1, 1)-design is a linear space where each point has degree n + 1. It is easy to prove that finite projective and affine planes are (n + 1, 1)-designs. By Π n we denote a finite projective plane of order n. In Π n every point is incident with n + 1 lines, every line is incident with n + 1 points and the plane contains n 2 + n + 1 lines and n 2 + n + 1 points. An affine plane of order n contains n 2 points and n 2 + n lines. Through each point there pass n + 1 lines and each line contains n points Special point sets in finite projective and affine planes Definition Let S be a point set in Π q. A line l is said to be a tangent to S at the point P if S l = {P }. If l meets S in k > 1 points, then we say that l is a k-secant of S. Lines not intersecing S are called skew to S. Blocking sets, covers Definition A blocking set B in a projective or an affine plane of order q is a set of points which intersects every line in the plane. If B contains a line, then it is called trivial. A point P in a blocking set B is essential if B \ {P } is not a blocking set, i.e. there is a tangent line to B through P. A blocking set B is said to be minimal if no proper subset of B is a blocking set or, equivalently, each of its points is essential. Definition If l is a line containing at most q points of a non-trivial minimal blocking set B, then B q + l B. In case of equality, B is a blocking set of Rédei type and l is a Rédei line of B. Definition A t-fold blocking set (or double blocking set when t = 2) in a projective or an affine plane is a point set B which intersects every line of the plane in at least t points. A point P B is essential in B if there is a t-secant of B passing through P. The t-blocking set B is said to be minimal if each of its points is essential. Definition A cover of Π q with h holes is a set of lines L that covers all but h points in Π q. Covers with one hole are the same as affine blocking sets in the dual plane. Some deep results on blocking sets have been collected in Section 2.4. Arcs, ovals, (k, n)-arcs, subplanes and unitals 8

14 Definition A k-arc in Π q is a set of k points such that no three of them are collinear. The (q + 1)-arcs are called ovals, the (q + 2)-arcs are called hyperovals. Definition A (k, n)-arc in Π q is a point set K of size k such that no line intersects K in more than n points. Definition A set of points and lines in Π q is called a subplane if they satisfy the three axioms in Definition Subplanes of order q are called Baer subplanes. For a line l of Π q we say that l is in Π s Π q if l intersects Π s in s + 1 points. Definition A unital in Π q, q square, is a set of q q + 1 points intersecting each line in 1 or q + 1 points. The index and the type of a point set Definition The index of a point set S in Π q is the smallest integer n such that S is contained in the union of n lines. Definition A point set S in Π q is of type (k 1, k 2,..., k l ) if for each line l, there is an index 1 i l such that S l = k i. Theorem (Tallini Scafati [84, 85]). If q is a power of a prime, then each point set of type (1, n) is one of the following: a unique point, a line, a Baer subplane or a unital. For other results on point sets of type (1, n) see [13, 99]. There are several results on (q + t, t)-arcs of type (0, 2, t), t 0, 2. Korchmáros and Mazzocca [66] proved that (q + t, t)-arcs of type (0, 2, t) exist in PG(2, q) only if q is even and t q. They also provided infinite families of examples whenever GF(q/t) is a subfield of GF(q). If T is a (q + t, t)-arc of type (0, 2, t), then it is easy to see that through each point of T there passes exactly one t-secant. In [45] new constructions were given by Gács and Weiner and they proved that the q/t + 1 t-secants of T pass through one point, called the t-nucleus of T (for t = 1, see [57, Lemma 8.6]). Recently Vandendriessche [101] found a new infinite family with t = q/ Projective and affine spaces over a finite field Let V = V (n + 1, q) denote the vector space of rank n + 1 over GF(q). The n- dimensional projective space over GF(q), denoted by PG(n, q), is the geometry whose k- dimensional (projective) subspaces are the k+1 dimensional subspaces of V. We denote 9

15 by a 1, a 2,..., a k the GF(q)-subspace of V spanned by the vectors a 1, a 2,..., a k V. We use this notation also for the respective projective subspace of PG(n, q). Note that the rank as a projective subspace is one less than the rank as a subspace of V. The 0, 1, 2 and n 1 dimensional subspaces of PG(n, q) are also called points, lines, planes and hyperplanes of PG(n, q). For any non-zero vector v V, the point v PG(n, q) can be represented by any non-zero vector w V such that v = w. Two non-zero vectors, v and w, span the same GF(q)-subspace in V if and only if v = λw for some λ GF(q). Thus the points of PG(n, q) can be represented by (n + 1)-tuples (v 1, v 2,..., v n+1 ) (0, 0,..., 0) that are well-defined up to a scalar multiplier. These (n + 1)-tuples are called homogeneous coordinates. If A is an n- dimensional subspace of V, then A = {v V : va T = 0} for a suitable non-zero vector a = (a 1, a 2,..., a n+1 ) V, where va T denotes the inner product of the vectors v and a T. As in the case of points, a is well-defined up to a scalar multiplier, hence we can represent A by the homogeneus coordinates of a. To distinguish hyperplanes and points, we use the notation [a 1, a 2,..., a n ] in the case of hyperplanes. The hyperplane with equation [0, 0, 1] in PG(2, q) will be called the line at infinity and will be denoted by l. The affine plane AG(2, q) will be usually considered as PG(2, q) \ l. The point set of AG(2, q) is the set of points of PG(2, q) \ l, the lines of AG(2, q) are the point sets (l \ l ), where l l and l is a line of PG(2, q). The affine coordinates of a point (a 1, a 2, 1) PG(2, q) \ l are (a 1, a 2 ). The points on the line l are called directions or ideal points, while the non-ideal points (lines) are called affine points (lines). For a point (1, m, 0) we also use the notation (m), where m is the slope of the affine lines through (1, m, 0). The point (0, 1, 0), that is the common point of vertical lines, will be denoted also by ( ). A finite projective plane Π q is isomorphic to PG(2, q) if and only if it satisfies Desargues Theorem. For this reason PG(2, q) is called the Desarguesian projective plane of order q. Note that if q < 9, then PG(2, q) is the unique projective plane of order q Associated polynomials Consider a subset U = {(a i, b i ) : i = 1, 2,..., U } of the affine plane AG(2, q). The (affine) Rédei polynomial of U is U H(X, Y ) = (X + a i Y b i ) = h j (Y )X U j, U i=1 j=0 10

16 where h j (Y ) is a polynomial of degree at most j in Y and h 0 (Y ) 1. Let H m (X) be the one-variable polynomial H(X, m) for any fixed value m. Then H m (X) GF(q)[X] is a fully reducible polynomial which reflects some geometric properties of U. Lemma (Folklore). Let H(X, Y ) be the Rédei polynomial of the point set U, and let m GF(q). Then X = k is a root of H m (X) with multiplicity r if and only if the line with equation Y = mx + k meets U in exactly r points. We need another result on polynomials. It has been developed by Szőnyi and Weiner in [92, 94, 102, 103]. For its proof we also recommend the Appendix in Héger s PhD Thesis [54] or [96] by Sziklai. For r R, let r + = max{0, r}. Theorem (Szőnyi Weiner Lemma, [54, 94, 96]). Let f, g GF(q)[X, Y ] be twovariable polynomials. Let d = deg f and suppose that the coefficient of X d in f is non-zero. For y GF(q), let m y = deg gcd (f(x, y), g(x, y)), where gcd denotes the greatest common divisor of the two polynomials in GF(q)[X]. Then for any y 0 GF(q), (m y m y0 ) + (deg f(x, Y ) m y0 )(deg g(x, Y ) m y0 ). y GF(q) The Hasse Bound and the Lang Weil estimate Definition Let f F[X 0, X 1,..., X n ] be a homogeneus polynomial. Then the point set F = V(f) = {(x 0, x 1,..., x n ) PG(n, F): f(x 0, x 1,..., x n ) = 0}, is called a hypersurface. The degree of F is deg f. A hypersurface in PG(2, F) is an algebraic curve. If F = GF(q), then the points of F are called the GF(q)-rational points or rational points of F. Definition A polynomial f in F[X 1, X 2,..., X n ] is said to be absolutely irreducible if it is irreducible over the algebraic closure of F. The hypersurface V(f) is said to be (absolutely) irreducible if f is (absolutely) irreducible. Theorem (Hasse Bound [58, Section 9]). Let C be an irreducible cubic curve over GF(q) and denote by N q (C) the number of rational points of C. Then we have N q (C) (q + 1) q q. 11

17 Theorem (Lang-Weil [70]). Let F be an absolutely irreducible hypersurface in PG(n, q) with degree d and let N q (F) denote the number of rational points of F. Then we have N q (F) q n 1 (d 1)(d 2)q n 3/2 + Cq n 2, where C is a constant not depending on q Collineations Let S and S be two spaces PG(n, F). First assume n 2, then a collineation is a bijection δ from S to S which preserves incidence, that is, if Π r Π s are two projective subspaces of S, then for their images we have Π δ r Π δ s. Note that, if δ is a bijection preserving incidence between points and lines, then it extends uniquely to a collineation. If n = 1, then a collineation between the lines S and S is the restriction of a collineation δ between two planes H and H, where S H, S H and S δ = S. A projectivity is a bijection δ between S and S defined by a non-singular matrix M. The image of P = (x 1, x 2,..., x n+1 ) S is P δ = M(x 1, x 2,..., x n+1 ) T S. Note that a projectivity is a collineation. If S = S, then we say that δ is a collineation (resp. The group of collineations of PG(n, q) is denoted by PΓL(n + 1, q). projectivity) of PG(n, F). The group of projectivities of PG(n, q) is the projective general linear group denoted by PGL(n+1, q). Theorem ( The Fundamental Theorem of Projective Geometry [57, Section 2.1.2]) 1. If δ is a collineation of PG(n, F), then δ = σδ, where δ is a projectivity of PG(n, F) and σ is a collineation induced by an automorphism of F. 2. If {P 0, P 1,..., P n+1 } and {P 0, P 1,..., P n+1 } are two subsets of PG(n, F) of size n + 2 such that no n + 1 points chosen from the same set lie in a hyperplane, then there exists a unique projectivity δ such that Pi δ = P i for all i {0, 1,..., n + 1}. Consider GF(q n ) as an n-dimensional vector space over GF(q). We can identify the m-dimensional GF(q)-subspaces with the (m 1)-dimensional projective subspaces of PG(n 1, q). For x, y GF(q n ) the two points x, y PG(n 1, q) are the same if and only if x q 1 = y q 1. Three points x, y, z are collinear if and only if there exist a, b, c GF(q), not all zero, such that ax + by + cz = 0. Let α be a primitive element of GF(q n ). The collineation group generated by x αx is a Singer group G of PG(n 1, q), that is, a cyclic collineation group of order (q n 1)/(q 1) permuting 12

18 the points in one orbit. If n = 2m + 2, then there is a subgroup H of order q m in the group G and Ebert [39] proved that if m is odd, then the orbits of H are caps, i.e. a set of points of PG(n 1, q) no three of which are collinear. For a short proof see Szőnyi [90]. Cossidente and Storme proved that these caps are the intersection of m linearly independent elliptic quadrics [26]. We will use this construction in PG(3, q), that is, when m = 1. Let Q = { x PG(3, q): x GF(q 4 ), x (q2 +1)(q 1) = 1}, that is the orbit of H containing 1. In this special case it can be seen easily that Q is an elliptic quadric. Proposition (Cossidente Storme [26, Theorem 3.7]). The cap Q = { x PG(3, q): x GF(q 4 ), x (q2 +1)(q 1) = 1} is an elliptic quadric in PG(3, q). Proof. Define Q(x) : GF(q 4 ) GF(q) as Q(x) = ε(x q3 +q x q2 +1 ), where ε q 1 = 1. We show that Q(x) is a quadratic form. Then it follows that { x PG(3, q): x GF(q 4 ), Q(x) = 0} = Q is a quadric. We have to show that (1) Q(λu) = λ 2 Q(u) for all λ GF(q) and u GF(q 4 ), and (2) B Q (u, v) := Q(u + v) Q(u) Q(v) is symmetric and bilinear. (1) trivially holds and we have B Q (u, v) = ε(v q3 u q + v q u q3 vu q2 v q2 u) that is symmetric and bilinear. The size of Q is q 2 + 1, thus Q(x) is non-degenerate and it defines an elliptic quadric. 2.4 Results on minimal blocking sets Minimal blocking sets play an important role in our work. We collect some of the most important results about these objects. For the definitions see Section The first result we cite is the oldest one. It is also called as the Bruen Pelikán theorem. Theorem (Bruen [1]). A non-trivial blocking set B in Π q has at least q + q + 1 points. In the case of equality B is a Baer subplane. Theorem (Blokhuis [15]). If B is a minimal non-trivial blocking set in PG(2, p), p > 2 prime, then B 3(p + 1)/2. In the case of equality there pass exactly (p 1)/2 tangent lines through each point of B. 13

19 Example ([57, Lemma 13.6]). Denote by C the set of non-zero squares in GF(q), q odd, and let S t = {(c, 0, 1), (0, c, 1), (c, 1, 0): c C} {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. This point set is called projective triangle and it shows the sharpness of Theorem Definition A blocking set B in PG(2, q) is called small if B < 3(q + 1)/2. Theorem (Szőnyi [91]). If B is a blocking set in PG(2, q), q = p h, p prime, of size at most 2q, then B contains a unique minimal blocking set. If B is a small minimal blocking set, then each line intersects B in 1 (mod p e ) points for some integer e 1. Theorem (Szőnyi [91]). If B is a small minimal blocking set in PG(2, p 2 ), p prime, then B is a line or a Baer subplane. Theorem provides lower and upper bounds for the possible sizes of minimal blocking sets. The bounds in the next result are due to Blokhuis and Polverino [82]. An improvement on the intersections with lines is from [95]. The divisibility condition is due to Sziklai [98]. Theorem (Szőnyi [91] and Polverino [82] and Sziklai [98] and Blokhuis) Let B be a small minimal blocking set in PG(2, q), q = p h, p prime. Then there exists a positive integer e, called the exponent of B, such that e divides h and q/p q p e e + 1 p e + 1 B 1 + (pe + 1)(q + 1) D, 2 where D = (1 + (p e + 1)(q + 1)) 2 4(p e + 1)(q 2 + q + 1). If B lies in the interval belonging to e and p e 4, then each line intersects B in 1 (mod p e ) points. Theorem (Blokhuis Storme Szőnyi [19]). Let B be a minimal blocking set in PG(2, q), q = p 2d+1, p prime. Let c p = 2 1/3 for p = 2, 3 and c p = 1 for p > 3. If B < q c p q 2/3, then B is a line. Theorem can be generalized for t-fold blocking sets. Theorem (Blokhuis Lovász Storme Szőnyi [2]). In PG(2, q), a minimal t-fold blocking set of size less than tq + (q + 3)/2 intersects each line in t (mod p) points. 14

20 2.4.1 Rédei type blocking sets and the direction problem Definition Consider PG(2, q) as AG(2, q) l. Let U be a set of points of AG(2, q). A point P of l is called a direction determined by U if there is a line through P that contains at least two points of U. The set of directions determined by U is denoted by D U. If U = q, and ( ) / D U, then U can be considered as a graph of a function, and U D U is a blocking set of Rédei type. We will need results on the number of directions determined by a set of q affine points in AG(2, q), and results on the extendability of a set of almost q affine points to a set of q points such that the two point sets determine the same directions. Theorem (Rédei Megyesi [83]). If a set of p points in AG(2, p), p prime, is not a line, then it determines at least (p + 3)/2 directions. Theorem (Blokhuis Ball Brouwer Storme Szőnyi [18] and Ball [8]). Let U be the set of q affine points in AG(2, q), q = p h, p prime. Let z = p e be maximal having the property that if P D U and l is a line through P, then l intersects U in 0 (mod z) points. Then one of the following holds: 1. z = 1 and (q + 3)/2 D U q + 1, 2. GF(z) is a subfield of GF(q) and q/z + 1 D U (q 1)/(z 1), 3. z = q and D U = 1. Theorem (Lovász Schrijver [74]). Let U be the set of p affine points in AG(2, p), p prime. If D U = (p + 3)/2, then U is affinely equivalent to the graph of the function x x p+1 2. Equivalently, if B is a blocking set of Rédei type of size 3(q + 1)/2 in PG(2, p), p prime, then B is projectively equivalent to the projective triangle. Theorem (Gács [43]). Let U be the set of p affine points in AG(2, p), p prime. If D U > (p + 3)/2, then D U [2 p ]. Theorem (Gács Lovász Szőnyi [46]). Let U be the set of p 2 affine points in AG(2, p 2 ), p prime. If D U (p 2 + 3)/2, then D U = (p 2 + 3)/2 and U is affinely equivalent to the graph of the function x x p or D U (p 2 + p)/ As a consequence, if B is a blocking set of Rédei type of size 3(p 2 +1)/2 in PG(2, p 2 ), p prime, then B is projectively equivalent to the projective triangle. 15

21 The first theorem on extendability was proved by Blokhuis [14, Proposition 2]. See also [89, Remark 7] by Szőnyi. Theorem (Blokhuis [14]). Let U AG(2, q) be a point set of size q 1. Then there exists a unique point P such that the point set U {P } determines the same directions as U. Later on, Sziklai [97] slightly extended the results of Szőnyi [89] and proved the following. Theorem (Sziklai [97]). Let U AG(2, q) be a point set of size q n, where n α q for some 1/2 α < 1. If D U < (q + 1)(1 α), then U can be extended to a set U of size q such that U determines the same directions as U Affine blocking sets, covers Theorem (Jamison [60], Brouwer-Schrijver [23]). If B is an affine blocking set of AG(2, q), then B 2q 1. As the dual of the above statement we get the following. If L is a set of lines of PG(2, q) that covers all points except one (which is not covered), then L 2q 1. In [21] the authors proved the following easy but very useful result which can be shown by adding at most h/2 lines and ending up with a cover having precisely one hole. Proposition (Blokhuis Brouwer Szőnyi [21, Proposition 1.5]). A partial cover of PG(2, q) with h > 0 holes, not all on one line, has size at least 2q 1 h/2. Proposition (Blokhuis Brouwer Szőnyi [21, Proposition 1.6]). A cover of the complement of a conic in PG(2, q), q odd, by elliptic lines contains at least 3(q 1)/2 lines. This is best possible for q = 3, 5, 7, 11 and there is no other example of size 3(q 1)/2 for odd q, q < Perspective point sets Definition Let l 1 and l 2 be two lines in a projective plane and let P denote their common point. We say that X 1 l 1 \P and X 2 l 2 \P are two perspective point sets if there is a point Q such that each line through Q intersects both X 1 and X 2 or intersects none of them. In other words, there is a perspectivity which maps X 1 onto X 2. 16

22 The following theorem characterizes perspective point sets in PG(2, q). This result was first published by Korchmáros and Mazzocca in [67] but we will use the notation of [24] by Bruen, Mazzocca and Polverino. Theorem (Korchmáros Mazzocca [67] and Bruen Mazzocca Polverino [24, Result 2.2, Result 2.3 and Result 2.4]). Let l 1 and l 2 be two lines in PG(2, q), q = p r, and let P denote their common point. Let X 1 l 1 \ P and X 2 l 2 \ P be two perspective point sets. Denote by U the set of all points which are centers of a perspectivity mapping X 1 onto X 2. Using a suitable projective frame in PG(2, q), there exist an additive subgroup B of GF(q) and a multiplicative subgroup A of GF(q) such that: (a) B is a subspace of GF(q) of dimension h 1 considered as a vectorspace over a subfield GF(q 1 ) of GF(q) with q 1 = p d and d r. This implies that B is an additive subgroup of GF(q) of order p h with h = dh 1. (b) A is a multiplicative subgroup of GF(q 1 ) of order n, where n (p d 1). In this way, B is invariant under A, i.e. B = AB := {ab : a A, b B}. (c) If G i denotes the full group of affinities of l i \ P preserving the set X i, i = 1, 2, then G 1 = G2 = G = G(A, B) = {g : g(y) = ay + b, a A, b B} Σ, where Σ is the full affine group on AG(1, q). (d) X i is a union of orbits of G i on l i \ P, i = 1, 2, and U = G = np h. (e) For every two integers n, h, such that n (p d 1) and d gcd(r, h), there exists in Σ a subgroup of type G = G(A, B) of order np h, where A and B are multiplicative and additive subgroups of GF(q) of order n and p h, respectively. (f) G has one orbit of length p h on AG(1, q), namely B, and G acts regularly on the remaining orbits, say O 1, O 2,..., O m, where m = q ph np h = pr h 1. n In the sequel we denote by B i the orbit of G i on l i \ P corresponding to B and by O1 i, Oi 2,..., Oi m the remaining orbits, for i = 1, 2. With this notation B 1 is the image of B 2 under the perspectivities with centers in U and also Oj 2 is the image of O1 j for j = 1, 2,..., m and vice versa. (g) B 1 X 1 if and only if B 2 X 2 and the same holds for the other orbits, i.e. Oj 1 X 1 if and only if Oj 2 X 2, for j = 1, 2,..., m. 17

23 (h) If a line l not through P meets U in at least two points, then l intersects both B 1 and B 2. Exactly one of the following cases must occur. 1. Both A and B are trivial. Then U consists of a singleton. 2. A is trivial and B is not trivial. Then U is a set of p h points all collinear with P. 3. B is trivial and A is not trivial. Then U is a set of n points on a line not through P. 4. A and B are the multiplicative and the additive group, respectively, of a subfield GF(p h ) of GF(q). Then U B 1 B 2 {P } = PG(2, p h ). 5. None of the previous cases occur. Then U is a point set of size np h and of type (0, 1, n, p h ), i.e. 0, 1, n, p h are the only intersection numbers of U with respect to the lines in PG(2, q). In addition, using the fact that U = np h, there are exactly n lines intersecting U in exactly p h points and they are all concurrent at the common point P of l 1 and l 2, each line intersecting U in exactly n points meets both B 1 and B 2. Proposition ([24, pg ]). Suppose that in Theorem the coordinate frames have been chosen such that l 1 = [0, 1, 0], l 2 = [0, 0, 1] and (0, 1, 1) U. Then we have the following: B 1 = {(b, 0, 1): b B} and B 2 = {(b, 1, 0): b B}, Oi 1 = {(ac i + b, 0, 1): a A, b B} and Oi 2 = {(ac i + b, 1, 0): a A, b B}, where c i is a suitable element of GF(q) \ B, for i = 1, 2,..., m, U = {(b, a, 1): a A, b B}. 2.6 Results from additive combinatorics By Theorem 2.3.7, the group PGL(3, q) of projective linear transformations of PG(2, q) is transitive on quadrangles. Thus every triangle T can be mapped by an element of 18

24 PGL(3, q) to the triangle with sides [1, 0, 0], [0, 1, 0] and [0, 0, 1] (and we can prescribe the image of another point not in T ). Similarly, every three concurrent lines can be mapped to [1, 0, 0], [1, 0, 1] and [0, 0, 1], that are lines with the common point (0, 1, 0). The next result shows how can we translate combinatorial problems on point sets of index three in PG(2, q) into problems in GF(q) + or in GF(q). Theorem (Folklore). If x, y, z GF(q), then the points (0, y, 1), (x, 0, 1) and (1, z, 0) are collinear if and only if y = xz. If x, y, z GF(q), then the points (0, x, 1), (1, y, 1) and (1, z, 0) are collinear if and only if y = x + z. Proof. Three points in PG(2, q) are collinear if and only if they are linearly dependent as vectors in V (3, q), i.e. when the determinant of the corresponding 3 3 matrix is zero. In the literature of difference sets, groups are usually written multiplicatively, but we will use difference sets only in abelian groups, mostly in GF(q) +, thus we give the related results and definitions such that we write groups additively. Some of the following notions will be used also in GF(q), in this case we will always write the group multiplicatively, while in the case of GF(q) + we will always use the additive notation. Definition Let G be a finite group (written additively). For a subset D of G we define D as the set { d: d D}. If A and B are finite non-empty subsets of G, then by A + B we mean the set {a + b: a A, b B}. For x G let r A,B (x) := {(a, b) A B : a + b = x}. An equivalent definition is the following: r A,B (x) := (x A) B = (x B) A. (2.1) Definition Let A and B be finite, non-empty subsets of the finite abelian group G (written additively). The set of t-representable sums in A + B is A + t B := {x A + B : r A,B (x) t}. (Also N t (A, B) is used in the literature.) By Stab(A) := {h G: A + h = A} we denote the stabilizer of A (there are several other notions for this, such as symmetry or period ). It is easy to see that Stab(A) is a subgroup of G and A is the union of cosets of Stab(A). Proposition (Folklore). Let G be an abelian group (written additively), and A and B finite subgroups of G. Then A + B and A B are subgroups of G and we have 19

25 A + B A B = A B. If A and B are cosets of A and B, respectively, then A + B is a coset of A + B. If A B, then A B is a coset of A B. Corollary Let G be an abelian group (written additively), and A and B finite subgroups of G. If A and B are cosets of A and B, respectively, then for each x A + B we have r A,B (x) = A B / A + B. Proof. We use (2.1), i.e. r A,B (x) = (x A ) B. Then (x A ) and B are cosets of A and B respectively. Their intersection is not empty since x A +B. Thus Proposition yields (x A ) B = A B = A B / A + B = A B / A + B. Theorem (Kneser [65]). Let G be an abelian group, and A and B finite non-empty subsets of G. Then either A + B A + B or A + B = A + H + B + H H A + B H, where H = Stab(A + B). Theorem (Pollard [81]). Let G be the cyclic group of order p, p prime. Let A and B be finite non-empty subsets of G, then for every 1 d min{ A, B } the following holds. d A + i B d min{p, A + B d}. i=1 The next result is from [51], Chapter 12. A slightly weaker version was published in [50]. Also, Hamidoune and Serra have similar unpublished results, see [53]. Theorem (Grynkiewicz [50, 51]). Let G be an abelian group, d 1 be a positive integer and A, B G finite, non-empty subsets of G with A, B d. Then either d A + i B d( A + B ) 2d 2 + 3d 2, (2.2) i=1 or else there exist subsets A A and B B with l := A \ A + B \ B d 1, (2.3) A + d B = A + B = A + d B, (2.4) d A + i B d( A + B ) (d l)( H ρ) dl d( A + B H ), (2.5) i=1 20

26 where H = Stab(A + d B) and ρ = A + H A + B + H B. Lemma Suppose that (2.2) in Theorem fails for a positive integer d. Then we have H A + B A + d B d + 1. Proof. If (2.2) fails, then there exist subsets A A and B B such that A + B = A + d B. Let l = A \ A + B \ B. Applying Kneser s Theorem for the subsets A and B, we get H A + B A + B = A A \ A + B B \ B A + d B = A + B l A + d B. From (2.3) we have l d 1 and this finishes the proof. 2.7 Difference sets Definition Let G be a finite group (written additively) of order v. A k-element subset D of G is called a (v, k, λ)-difference set in G if r D, D (x) = λ for each x G\{0}. If G is cyclic (resp. abelian), then D is called a cyclic (resp. abelian) difference set. If λ = 1, then D is called a planar difference set. The complementary difference set of a (v, k, λ)-difference set D is D := G\D, which is a a (v, v k, v 2k +λ)-difference set. Definition Let D be a difference set in the finite group G (written additively). If G is the disjoint union of D, D and {0}, then D is called a skew Hadamard difference set. Example (Paley difference set). Let D be the set of non-zero squares in GF(q), where q is a prime power congruent to 3 modulo 4. Then D is a skew Hadamard difference set in GF(q) + with parameters (q, (q 1)/2, (q 3)/4). It is conjectured that if an abelian group G contains a skew Hadamard difference set, then G is elementary abelian. If D is a skew Hadamard difference set, then its parameters satisfy (v, k, λ) = (4λ + 3, 2λ + 1, λ). For a short survey on skew Hadamard difference sets see the introduction of [42] by Feng and Xiang. We will need the following easy result. Theorem (Coulter Gutekunst [27, Theorem 3.1]). If D is a skew Hadamard difference set in G with parameters (v, k, λ), then r D,D (x) = λ for each x D and r D,D (y) = λ + 1 for each y D. Definition Let G be a finite group (written additively) of order v. A k-element subset D of G is called a (v, k, λ, µ) partial difference set (PDS) if r D, D (x) = λ for each x D \ {0} and r D, D (y) = µ for each y G \ (D {0}). 21

27 A PDS with λ = µ is a difference set, otherwise we have the following result by Ma. Theorem (Ma [76, Theorem]). If D is a (v, k, λ, µ)-pds with λ µ, then D = D. If D is a PDS with D = D, then D {0}, D \ {0}, G \ D, G \ (D {0}) and (G \ D) {0} are partial difference sets. For a survey on partial difference sets see [75] by Ma, where one can find many examples in elementary abelian p-groups. We will need these objects exactly in these groups. 22

28 Chapter 3 Semiarcs in finite projective planes 3.1 Semiovals Definition Let Π q be a projective plane of order q. A semioval in Π q is a nonempty point set S with the property that for every point P S there exists a unique line l, such that S l = {P }. This line is called the tangent to S at P. The definition of semiovals is due to Buekenhout [25]. He also defined their higher dimensional analogues: a semiovoid O is a non-empty point set in PG(n, q), n > 2, such that for all P O the union of all tangents to O at P is a hyperplane. In [100] Thas proved the following conjecture of Buekenhout: the only semiovoids in PG(3, q) are the ovoids (point sets of size q such that no three of them are collinear) and there are no semiovoids in PG(n, q), n > 3. He also gave upper and lower bounds on the size of a semioval in Π q. Theorem (Thas [100]). If S is a semioval in a finite projective plane of order q, then q + 1 S q q + 1. It is easy to see that the size of S reaches the lower (upper) bound if and only if S is an oval (unital). The second smallest examples were constructed first by Blokhuis [14]. Example (Blokhuis [14]). Let S := {(0, 1, s), (s, 0, 1), (1, s, 0): s / C}, where C denotes the set of squares in GF(q). The point set S is a semioval of size 3(q 1)/2 in PG(2, q). (To see this, one should use Theorem ) 23

29 Seminuclear sets, almost oval semiovals Definition An almost oval semioval is a point set S of size q+a in Π q such that through each point of S there pass exactly one tangent and exactly one (a + 1)-secant. When a = 2, then these point sets were called seminuclear sets by Blokhuis and Bruen [16]. The name almost oval is from the book of Sziklai [96]. In [14], Blokhuis proved the following theorem. Theorem (Blokhuis [14]). If S is an almost oval semioval in PG(2, q), then S is as in Example or it is the symmetric difference of two lines with one further point removed from both lines. Each semioval of size q + 2 is an almost oval semioval, hence the above theorem has the following corollary. Corollary (Blokhuis [14]). If S is a semioval of size q + 2 in PG(2, q), then q = 7 and S is as in Example or q = 4 and S is the symmetric difference of two lines with one further point removed from both lines. Blocking semiovals Definition A blocking semioval is a semioval that is also a blocking set. Note that a blocking semioval S is necessarily a minimal blocking set and a maximal semioval, that is, S cannot be contained in any semioval properly. The study of blocking semiovals was suggested by Batten. She invented a cryptographic protocol which uses determining sets, and blocking semiovals are determining sets in projective planes, see [9]. Blocking semiovals have been constructed by many authors, see for example [38] by Dover, Mellinger and Wantz where the authors construct blocking semiovals containing conics. For other constructions related to semiovals and conics, see [37, 64, 73, 88]. We cite here some recent results on the sizes of blocking semiovals. The first result is by Dover, the second one is due to Héger and Takáts. Theorem (Dover [36]). If S is a blocking semioval in Π q, q 7, then S 2q + 2q 47/4 1/2. Theorem (Héger-Takáts [55]). If S is a blocking semioval in PG(2, q), q 4, then S 9q/4 3. Regular semiovals 24