Robert S. Coulter

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Planar functions were introduced by Dembowski and Ostrom ([4]) 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 presemifields 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)
where f ∈ Fq[X ]. These sums are also known as Weil sums. The problem of explicitly evaluating these sums is quite often difficult. Results giving estimates for the absolute value of the sum are more common and such results have been regularly appearing for many years. The book [5] by Lidl and Niederreiter gives an overview of this area of research in the(More)
BACKGROUND The purpose of this study was to fill a significant void in the ophthalmic literature by performing a large scale, comprehensive, prospective study of the prevalence of vision disorders and ocular pathology in a clinical pediatric population using well-defined diagnostic criteria. METHODS A prospective study was performed on 2,023 consecutive(More)
where Fq denotes the finite field of q elements (q = p e for p a prime) and f ∈ Fq[X]. In a recent article [2] the author gave explicit evaluations of all Weil sums with f(X) = aX α+1 and p odd. These were obtained mostly through generalising methods used by Carlitz in [1] who obtained explicit evaluations of Weil sums with f(X) = aX + bX, p odd. In this(More)
Let Fq denote the finite field with q = p e elements and P(Fq ) be the n-dimensional projective space over Fq . For any f ∈ Fq [X1, . . . , Xn] of degree d, define the homogenous polynomial f ∗ ∈ Fq[X0, . . . , Xn] by f∗(X0, . . . , Xn) = X d 0f(X1/X0, . . . , Xn/X0). The set of Fq -rational points of an algebraic hypersurface Xf is the set of all points P(More)
We report on a recent implementation of Giesbrecht’s algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing p-polynomials over a finite field, and how Giesbrecht’s algorithm is outlined in some detail by Ore in the 1930’s. We end with some(More)
Abstract. The definition of bent is redefined for any finite field. Our main result is a complete description of the relationship between bent polynomials and perfect non-linear functions over finite fields: we show they are equivalent. This result shows that bent polynomials can also be viewed as the generalisation to several variables of the class of(More)