Rex W. Matthews

Planar Functions and Planes of Lenz-Barlotti Class II

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 question posed in [4]. These planar functions define at least one such affine plane of order 3e for every e ≥ 4 and their projective closures are of Lenz-Barlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner.

Permutation polynomials in one and several variables

Various authors have dealt with problems relating to permutation polynomials over finite systems ([4], [8], [10], [18], [20]-[25],[29]-[33], etc.). In this thesis various known results are extended and several questions are resolved. Chapter 2 begins by considering the problem of finding those permutation polynomials in a single variable amongst some given classes of polynomials. Previously, this question was settled only for cyclic polynomials and Chebyshev polynomials of the first kind. Here we consider the Chebyshev polynomials of the second kind and polynomials of the form (x n- 1)/(x - I). Certain questions on multivariable polynomials are then considered. Chapter 3 deals with questions involving polynomials whose coefficients lie in a subfield of the given field, and considers some combinatorial questions. Chapter 4 resolves the structure of the group of maps of F nq → F nq induced by the extended Chebyshev polynomials of Lidl and Wells [26]. Chapter 5 extends this further to finite rings -Z/(pe), thus generalising results of Lausch-Műller-Nőbauer [18]. Chapter 6 settles some questions concerning the conjecture of Schur on polynomials f(x) ϵ Z[x] which permute infinitely many residue fields Fp. It is known ([10]) that these are compositions of cyclic and Chebyshev polynomials of the first kind. In chapter 6 it is determined which of these polynomials have the required property.

A note on constructing permutation polynomials

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 bijective 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.

Bent polynomials over finite fields

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 polynomials known as planar polynomials. An explicit method for obtaining large sets of not necessarily distinct maximal orthogonal systems using bent polynomials is given and we end with a short discussion on the existence of bent polynomials over finite fields.

The structure of the group of permutations induced by Chebyshev polynomial vectors over the ring of integers mod m

In an earlier paper the author investigated the properties of a class of multivanable polynomial vectors which generalise the multivariable Chebyshev polynomial vectors. In this paper the behaviour of these polynomials over rings of the type Z/(m) is investigated, and conditions are determined for such an n -variable polynomial vector to induce a permutation of (Z/(m)) n . More detailed results on the Chebyshev polynomial vectors follow. The composition properties of these vectors imply that the permutations induced by certain subsets of them form groups under composition of mappings, and the structure of these groups is investigated.

On the number of distinct values of a class of functions over a finite field

Several authors have recently shown that a planar function over a finite field of order q must have at least (q+1)/2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial @f(X,Y) can be written as f(X+Y)-f(X)-f(Y) for some polynomial f. Using the ideas of the algorithm, we then show a Dembowski-Ostrom polynomial is planar over a finite field of order q if and only if it yields exactly (q+1)/2 distinct values under evaluation; that is, it meets the lower bound of the image size of a planar function.

Planar polynomials and an extremal problem of Fischer and Matoušek

Abstract Let G be a 3-partite graph with k vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matousek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with ( 1 − o ( 1 ) ) k 3 / 2 triangles, and a double counting argument shows that one cannot have more than ( 1 + o ( 1 ) ) k 7 / 4 triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to ( 1 − o ( 1 ) ) k 5 / 3 .