115 research outputs found
Finite semifields with a large nucleus and higher secant varieties to Segre varieties
In [2] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [7] we proved that the configuration needed for the geometric construction given in [2] for finite semifields is equivalent with an (n - 1)-dimensional subspace skew to a determinantal hypersurface in PG (n(2) - 1, q), and provided an answer to the isotopism problem in [2]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre
Subgeometries and linear sets on a projective line
We define the splash of a subgeometry on a projective line, extending the
definition of \cite{BaJa13} to general dimension and prove that a splash is
always a linear set. We also prove the converse: each linear set on a
projective line is the splash of some subgeometry. Therefore an alternative
description of linear sets on a projective line is obtained. We introduce the
notion of a club of rank , generalizing the definition from \cite{FaSz2006},
and show that clubs correspond to tangent splashes. We determine the condition
for a splash to be a scattered linear set and give a characterization of clubs,
or equivalently of tangent splashes. We also investigate the equivalence
problem for tangent splashes and determine a necessary and sufficient condition
for two tangent splashes to be (projectively) equivalent
Semifields from skew polynomial rings
Skew polynomial rings were used to construct finite semifields by Petit in
1966, following from a construction of Ore and Jacobson of associative division
algebras. In 1989 Jha and Johnson 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 2008 and implicitly in recent work by
Dempwolff
On BEL-configurations and finite semifields
The BEL-construction for finite semifields was introduced in \cite{BEL2007};
a geometric method for constructing semifield spreads, using so-called
BEL-configurations in . In this paper we investigate this construction
in greater detail, and determine an explicit multiplication for the semifield
associated with a BEL-configuration in , extending the results from
\cite{BEL2007}, where this was obtained only for . Given a
BEL-configuration with associated semifields spread , we also show
how to find a BEL-configuration corresponding to the dual spread
. Furthermore, we study the effect of polarities in on
BEL-configurations, leading to a characterisation of BEL-configurations
associated to symplectic semifields.
We give precise conditions for when two BEL-configurations in
define isotopic semifields. We define operations which preserve the BEL
property, and show how non-isotopic semifields can be equivalent under this
operation. We also define an extension of the ```switching'' operation on
BEL-configurations in introduced in \cite{BEL2007}, which, together
with the transpose operation, leads to a group of order acting on
BEL-configurations
Finite semifields and nonsingular tensors
In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)
Field reduction and linear sets in finite geometry
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 formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
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