Planar functions were introduced by Dembowski and Ostrom () to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a… (More)
Strong conditions are derived for when two commutative presemi-fields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely , any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all… (More)
We consider a class of Weil sums involving polynomials of a particular shape. In all cases, explicit evaluations are obtained.
We determine the number of Fq-rational points of a class of Artin-Schreier curves by using recent results concerning evaluations of some exponential sums. In particular, we determine infinitely many new examples of maximal and minimal plane curves in the context of the Hasse-Weil bound.
1 Dembowski-Ostrom Polynomials and Linearised Polynomials Let p be a prime and q = p e. Let F q denote the finite field of order q and F * q represent the set of non-zero elements of F q. The ring of polynomials in the indeterminate X with coefficients from F q will be represented by F q [X]. A polynomial f ∈ F q [X] which permutes F q under evaluation is… (More)
Planar functions were introduced by Dembowski and Ostrom in  to describe affine planes possessing collineation groups with particular properties. To date their classification has only been resolved for functions over fields of prime order. In this article we classify planar monomials over fields of order p 2 with p a prime. Let p be a prime, e a natural… (More)
Let H be a subgroup of the multiplicative group of a finite field. In this note we give a method for constructing permutation polynomials over the field using a bi-jective map from H to a coset of H. A similar, but inequivalent, method for lifting permutation behaviour of a polynomial to an extension field is also given.
A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.