BLOCKING SETS OF HALL PLANES, AND VALUE SETS OF POLYNOMIALS OVER - PowerPoint PPT Presentation

BLOCKING SETS OF HALL PLANES, AND VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Fq13, Gaeta June 5, 2017 jan@debeule.eu 1

This is joint work with ◮ Tamás Héger ◮ Tamás Szőnyi ◮ Geertrui Van de Voorde 2

Value sets VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Definition Let F q be the finite field of order q and f ( x ) ∈ F q [ x ] . Then V ( f ) := { f ( x ) | x ∈ F q } 3

Value sets VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Definition Let F q be the finite field of order q and f ( x ) ∈ F q [ x ] . Then V ( f ) := { f ( x ) | x ∈ F q } If | V ( f ) | = q , then f is a permutation polynomial 3

Value sets VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Observation: if f ( x ) has degree n then | V ( f ) | ≥ q n . 4

Value sets VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Observation: if f ( x ) has degree n then | V ( f ) | ≥ q n . Theorem (D. Wan, 1993) If f ( x ) has degree n and is not a permutation polynomial, then | V ( f ) | ≤ q − q − 1 . n 4

Value sets VALUE SETS OF PARTICULAR POLYNOMIALS Consider the following polynomial over F q h , h > 1, a ∈ Z : s a ( x ) := x a ( x + 1 ) q − 1 . Theorem (Cüsick and Müller, 1996) | V ( s 1 ) | = ( 1 − 1 q ) q h . q ) q h + 1 (Compare this with the result of Wan: | V ( s 1 ) | ≤ ( 1 − 1 q ). 5

Value sets VALUE SETS OF PARTICULAR POLYNOMIALS Theorem (Rosendahl 2008/9) If q ≡ 0 (mod 3 ) and h = 2 , then | V ( s 3 ) | = 2 3 q 2 − 1 6 q − 1 2 . 6

Blocking sets of projective planes PROJECTIVE PLANES A projective plane is a point-line geometry satisfying the following three axioms: ◮ Every two points determine exactly one line; ◮ Every two lines meet in exactly one point: ◮ There exist four points of which no three are collineair 7

Blocking sets of projective planes PROJECTIVE PLANES A projective plane is a point-line geometry satisfying the following three axioms: ◮ Every two points determine exactly one line; ◮ Every two lines meet in exactly one point: ◮ There exist four points of which no three are collineair Let K be any (skew)field, and V a 3-dimensional K vector space. A Desarguesian projective plane is the following structure: ◮ points are the 1-dimensional subspaces of V ; ◮ lines are the 2-dimensional subspaces of V ; ◮ incidence is symmetrised containment There exists many examples of non-Desarguesian projective planes. 7

Blocking sets of projective planes FINITE PROJECTIVE PLANES Let Π be a finite projective planes, i.e. with a finite number of points and lines. ◮ The natural number n ≥ 2 is the order of Π : every line is incident with n + 1 points, and every point is incident with n + 1 lines. ◮ There are exactly n 2 + n + 1 points and exactly the same amount of lines. ◮ A Desarguesian projective plane is always coordinatized over a skewfield, in the finite case this a field of order q , which is the order of the projective plane as well. 8

Blocking sets of projective planes BLOCKING SETS OF PROJECTIVE PLANES Let Π be a projective plane. Definition A blocking set is a set B of points of Π such that every line of Π meets B in at least one point. 9

Blocking sets of projective planes BLOCKING SETS OF PROJECTIVE PLANES Let Π be a projective plane. Definition A blocking set is a set B of points of Π such that every line of Π meets B in at least one point. The set of points on a line is an example. A blocking set containing a line is called trivial . 9

Blocking sets of projective planes BLOCKING SETS OF PROJECTIVE PLANES Let Π be a projective plane. Definition A blocking set is a set B of points of Π such that every line of Π meets B in at least one point. The set of points on a line is an example. A blocking set containing a line is called trivial . Definition A blocking set B is minimal for each point P ∈ B , there exists at least one line of Π meeting B only in P . 9

Blocking sets of projective planes BLOCKING SETS OF PROJECTIVE PLANES Theorem (Bruen, 1971) Let Π be a projective plane of order n . A non-trivial minimal blocking set contains at least n + √ n + 1 points. 10

Blocking sets of projective planes BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES Let Π q be a Desarguesian projective plane over the field F q , q = p h . Theorem (Blokhuis, 1994) Let q be prime, then a non-trivial minimal blocking set of Π q contains at least 3 ( q + 1 ) points. 2 11

Blocking sets of projective planes BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES Let Π q be a Desarguesian projective plane over the field F q , q = p h . Theorem (Blokhuis, 1994) Let q be prime, then a non-trivial minimal blocking set of Π q contains at least 3 ( q + 1 ) points. 2 Definition A blocking set of Π q is small if it contains less than 3 ( q + 1 ) points. 2 11

Blocking sets of projective planes BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES Let Π q be a Desarguesian projective plane over the field F q , q = p h . Theorem (Blokhuis, 1994) Let q be prime, then a non-trivial minimal blocking set of Π q contains at least 3 ( q + 1 ) points. 2 Definition A blocking set of Π q is small if it contains less than 3 ( q + 1 ) points. 2 Theorem (Szőnyi, 1997) If B is a small blocking set of Π q , q = p h , of size less than 3 ( q + 1 ) , 2 then each line meets B in 1 (mod p ) points. 11

Blocking sets of projective planes WHAT ABOUT NON-DESARGUESIAN PLANES ◮ What is small in a non-Desarguesian plane? ◮ What about the 1 mod p result? 12

Blocking sets of projective planes HALL PLANES 13

Blocking sets of projective planes MAIN IDEA TO CONSTRUCT A BLOCKING SET 14

Blocking sets of projective planes ONE OF THE MAIN LEMMA’S Lemma ◮ If q �≡ 2 (mod 3 ) , then ∀ P ∈ D ′ , t 0 ( P ) = q 2 − q . 3 ◮ If q ≡ 2 (mod 3 ) , then for q + 1 points P ∈ D ′ , 3 t 0 ( P ) = q 2 − q − 2 , and for 2 ( q + 1 ) points P ∈ D ′ , 3 3 t 0 ( P ) = q 2 − q + 1 . 3 15

Blocking sets of projective planes RESULTS FOR BLOCKING SETS Theorem In the projective Hall plane of order q 2 , q > 2 , there exists a minimal blocking set of size q 2 + 2 q + 2 , which admits 1 − , 2 − , 3 − , 4 − , ( q + 1 ) − and ( q + 2 ) -secants. Theorem Let q be a prime power. There exists a non-Desarguesian affine plane of order q 2 in which there is a blocking set of size at most 4 q 2 3 + 5 q 3 16

Blocking sets of projective planes RESULTS FOR BLOCKING SETS Theorem In the projective Hall plane of order q 2 , q > 2 , there exists a minimal blocking set of size q 2 + 2 q + 2 , which admits 1 − , 2 − , 3 − , 4 − , ( q + 1 ) − and ( q + 2 ) -secants. Theorem Let q be a prime power. There exists a non-Desarguesian affine plane of order q 2 in which there is a blocking set of size at most 4 q 2 3 + 5 q 3 Theorem (Jamison, 1977, Brouwer–Schrijver, 1978) A blocking set of a Desarguesian affine plane of order q has at least 2 q − 1 points. 16

Blocking sets of projective planes CONNECTION WITH VALUE SETS André planes: derivation in Desarguesian planes of order q h , h ≥ 2. 17

Blocking sets of projective planes CONNECTION WITH VALUE SETS André planes: derivation in Desarguesian planes of order q h , h ≥ 2. f a , c ( x ) := ax q − ac ∪ { ( 1 : ax q − 1 : 0 ) : x ∈ F ( q h ) ∗ } B ( a , c ) := { ( x : f a , c ( x ) : 1 ) : x ∈ F ( q h ) } . � �� � � �� � D ( a ) U ( a , c ) 17

Blocking sets of projective planes CONNECTION WITH VALUE SETS André planes: derivation in Desarguesian planes of order q h , h ≥ 2. f a , c ( x ) := ax q − ac ∪ { ( 1 : ax q − 1 : 0 ) : x ∈ F ( q h ) ∗ } B ( a , c ) := { ( x : f a , c ( x ) : 1 ) : x ∈ F ( q h ) } . � �� � � �� � D ( a ) U ( a , c ) sets B ( a , c ) , a ∈ D := { x q − 1 | x ∈ F ∗ q h }, c ∈ F q h are lines of type (ii) of the André plane. 17

Blocking sets of projective planes CONNECTION WITH VALUE SETS The set of points B 0 := { ( y : 1 : y q ) : y ∈ F ( q h ) } ∪ { ( 1 : 0 : y q − 1 ) : y ∈ F ( q h ) ∗ } � �� � � �� � U 0 D 0 is a blocking set (of Rédei - type) of the Desarguesian projective plane of order q h . 18

Blocking sets of projective planes CONNECTION WITH VALUE SETS The set of points B 0 := { ( y : 1 : y q ) : y ∈ F ( q h ) } ∪ { ( 1 : 0 : y q − 1 ) : y ∈ F ( q h ) ∗ } � �� � � �� � U 0 D 0 is a blocking set (of Rédei - type) of the Desarguesian projective plane of order q h . | V ( s − 1 ) | the number of lines of type (ii) of the André plane that are skew to the set U 0 . 18

Blocking sets of projective planes CONNECTION WITH VALUE SETS Theorem Let q be odd. The size of the value set of s − 1 ( x ) = x − 1 ( x + 1 ) q − 1 in F ( q 2 ) is  3 q 2 − 1 2 6 q − 1 if q �≡ 2 (mod 3 ) ,  2 | V ( s − 1 ) | = 3 q 2 − 1  2 6 q + 1 if q ≡ 2 (mod 3 ) . 6 19