Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In \cite{DoDu2008}, the intersection problem for subgeometries of $\PG(n,q)$ is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline $\PG(1,q)$ with a linear set in $\PG(1,q^h)$ and investigate the existence of {\em irregular} sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd $q$, on the size of the intersection of two different linear sets of rank 3 in $\PG(1,q^h)$
Based on the simple and well understood concept of subfields in a finite field, the technique called 'field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalised and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental questions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields.
A finite semifield is shown to be equivalent to the existence of a particular geometric configuration of subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field. In 1965 Knuth \cite{Knuth1965} showed that each finite semifield generates in total six (not necessarily isotopic) semifields. In certain cases, the geometric interpretation obtained here allows us to construct another six semifields, providing a link between some known examples which are not related by Knuth's operations
Abstract. Skew polynomial rings were used to construct finite semifields by Petit in [20], following from a construction of Ore and Jacobson of associative division algebras. Johnson and Jha [10] later constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in [13] and implicitly by Dempwolff in [2].
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