Polynomials in finite geometries

A method of using polynomials to describe objects in finite geometries is outlined and the problems where this method has led to a solution are surveyed. These problems concern nuclei, affine blocking sets, maximal arcs and unitals. In the case of nuclei these methods give lower bounds on the number of nuclei to a set of points in PG(n, q), usually dependent on some binomial coefficient not vanishing modulo the characteristic of the field. These lower bounds on nuclei lead directly to lower bounds on affine blocking sets with respect to lines. A short description of how linear polynomials can be used to construct maximal arcs in certain translation planes is included. A proof of the non-existence of maximal arcs in PG(2, q) when q is odd is outlined and some bounds are given as to when a (k, n)-arc can be extended to a maximal arc in PG(2, q). These methods can also be applied to unitals embedded in PG(2, q). One implication of this is that when q is the square of a prime a non-classical unital has a limited amount of Baer sublines amongst its secants.

[1]  R. C. Bose,et al.  Linear representations of projective planes in projective spaces , 1966 .

[2]  Aiden A. Bruen Polynomial Multiplicties over Finite Fields and Intersection Sets , 1992, J. Comb. Theory, Ser. A.

[3]  B. Segre Ovals In a Finite Projective Plane , 1955, Canadian Journal of Mathematics.

[4]  O. Ore On a special class of polynomials , 1933 .

[5]  Joseph A. Thas Some Results Concerning {(q + 1)(n - 1); n}-Arcs and {(q + 1)(n - 1) + 1; n}-Arcs in Fintie Projective Planes of Order q , 1975, J. Comb. Theory, Ser. A.

[6]  Udo Heim,et al.  Proper blocking sets in projective spaces , 1997, Discret. Math..

[7]  Aart Blokhuis,et al.  On multiple nuclei and a conjecture of Lunelli and Sce , 1994 .

[8]  Christine M. O'Keefe,et al.  Ovoids in PG(3, q): a survey , 1996, Discret. Math..

[9]  Nicholas A. Hamilton Some maximal arcs in Hall planes , 1995 .

[10]  Simeon Ball On Intersection Sets in Desarguesian Affine Spaces , 2000, Eur. J. Comb..

[11]  Simeon Ball Partial Unitals and Related Structures in Desarguesian Planes , 1998, Des. Codes Cryptogr..

[12]  R. Denniston Some maximal arcs in finite projective planes , 1969 .

[13]  Johannes André,et al.  Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe , 1954 .

[14]  Nicholas A. Hamilton Some inherited maximal arcs in derived dual translation planes , 1995 .

[15]  A. Barlotti,et al.  Some topics in finite geometrical structures , 1965 .

[16]  Simeon Ball,et al.  Multiple Blocking Sets and Arcs in Finite Planes , 1996 .

[17]  T Szönyi,et al.  Surveys in Combinatorics, 1997: Some Applications of Algebraic Curves in Finite Geometry and Combinatorics , 1997 .

[18]  Aart Blokhuis Polynomials in finite geometries and combinatorics , 1993 .

[19]  F. Buekenhout Existence of unitals in finite translation planes of order q2 with a kernel of order q , 1976 .

[20]  Joseph A. Thas,et al.  Construction of Maximal Arcs and Dual Ovals in Translation Planes , 1980, Eur. J. Comb..

[21]  Nicholas Hamilton,et al.  Some Maximal Arcs in Derived Dual Hall Planes , 1994, Eur. J. Comb..

[22]  Aiden A. Bruen,et al.  The Jamison method in galois geometries , 1991, Des. Codes Cryptogr..

[23]  Ha Henny Wilbrink,et al.  A characterization of exterior lines of certain sets of points in PG (2, q) , 1987 .

[24]  ROBERT E. JAMISON,et al.  Covering Finite Fields with Cosets of Subspaces , 1977, J. Comb. Theory, Ser. A.

[25]  Simeon Ball On Nuclei and Blocking Sets in Desarguesian Spaces , 1999, J. Comb. Theory, Ser. A.

[26]  Zoltán Füredi,et al.  Extremal problems in finite geometry , 1994 .

[27]  O. Ore Contributions to the theory of finite fields , 1934 .

[28]  C. T. Quinn,et al.  Concerning a characterisation of Buekenhout-Metz unitals , 1995 .

[29]  Alexander Schrijver,et al.  The Blocking Number of an Affine Space , 1978, J. Comb. Theory, Ser. A.

[30]  Simeon Ball,et al.  An easier proof of the maximal arcs conjecture , 1998 .

[31]  Tim Penttila,et al.  Sets of type (m, n) in the affine and projective planes of order nine , 1995, Des. Codes Cryptogr..

[32]  A. Blokhuis,et al.  On maximal sets of nuclei in PG(2,q) and quasi-odd sets in AG(2,q) , 1991 .

[33]  J. Thas Construction of maximal arcs and partial geometries , 1974 .

[34]  Aart Blokhuis On Nuclei and Affine Blocking Sets , 1994, J. Comb. Theory, Ser. A.

[35]  A. Blokhuis,et al.  On the Incompleteness of (k, n)-arcs in Desarguesian Planes of Order q Where n Divides q , 1999 .

[36]  Aart Blokhuis,et al.  Maximal arcs in Desarguesian planes of odd order do not exist , 1997, Comb..

[37]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[38]  J. Hirschfeld Projective Geometries Over Finite Fields , 1980 .

[39]  Francesco Mazzocca,et al.  Special point sets inPG(n,q) and the structure of sets with the maximal number of nuclei , 1991 .

[40]  Rudolf Metz On a class of unitals , 1979 .

[41]  Nicholas A. Hamilton,et al.  m-systems of polar spaces and maximal arcs in projective planes , 2000 .