On the number of directions determined by a pair of functions over a prime field

Authors:
Simeon Ball;András Gács;Peter Sziklai
Affiliations:
Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona 1-3, Mòdul C3, Campus Nord, 08034 Barcelona, Spain;Eötvös University Budapest, Pázmány P. sétány 1/c, Budapest, Hungary H-1117;Eötvös University Budapest, Pázmány P. sétány 1/c, Budapest, Hungary H-1117
Venue:
Journal of Combinatorial Theory Series A
Year:
2008

Citing 7
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Abstract

A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG(3,p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than (2@?p-16@?+1)(p+2@?p-16@?)/2~2p^2/9 pairs (a,b)@?F"p^2 with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e@?F"p with the property that f(x)=cg(x)+dx+e.