A Carlitz type result for linearized polynomials

For an arbitrary $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ we study the problem of finding those $q$-polynomials $g$ over $\mathbb{F}_{q^n}$ for which the image sets of $f(x)/x$ and $g(x)/x$ coincide. For $n\leq 5$ we provide sufficient and necessary conditions and then apply our result to study maximum scattered linear sets of $\mathrm{PG}(1,q^5)$

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

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

Compressed zero-divisor graphs of noncommutative rings

We extend the notion of the compressed zero-divisor graph $\varTheta(R)$ to noncommutative rings in a way that still induces a product preserving functor $\varTheta$ from the category of finite unital rings to the category of directed graphs. For a finite field $F$, we investigate the properties of $\varTheta(M_n(F))$, the graph of the matrix ring over $F$, and give a purely graph-theoretic characterization of this graph when $n \neq 3$. For $n \neq 2$ we prove that every graph automorphism of $\varTheta(M_n(F))$ is induced by a ring automorphism of $M_n(F)$. We also show that for finite unital rings $R$ and $S$, where $S$ is semisimple and has no homomorphic image isomorphic to a field, if $\varTheta(R) \cong \varTheta(S)$, then $R \cong S$. In particular, this holds if $S=M_n(F)$ with $n \neq 1$.Comment: 30 page