Fractional jumps: complete characterisation and an explicit infinite family

TL;DR

This paper characterizes transitive fractional jumps, showing they originate from projective automorphisms, and constructs an infinite family of such automorphisms using primitive polynomials for arbitrary dimensions.

Contribution

It provides a complete characterization of transitive fractional jumps and constructs an infinite family of such automorphisms for any dimension.

Findings

Transitive fractional jumps are derived from projective automorphisms.

An infinite class of projectively primitive polynomials is constructed.

Full orbit sequences can be generated over affine spaces for any dimension.

Abstract

In this paper we provide a complete characterisation of transitive fractional jumps by showing that they can only arise from transitive projective automorphisms. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space.