turkmath.org

In this talk we will concentrate on the geometric approach to the theory of finite semifields. We will explain the notion of a linear set in a projective space and elaborate on their use in the theory of finite semifields (see [4]), with an emphasis on the so-called scattered linear sets. These are linear sets which have maximal size with respect to their rank (see [1]). Next we will illustrate these techniques from finite geometry by addressing an isotopism problem posed by Dempwol in [2]. Dempwol gives a construction of three classes of rank two semields of order $q^{2n}$ , with $q$ and $n$ odd, using Dembowski-Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In [3] this problem was solved for $n > 3$, in particular it is proved that two of these classes, labeled $D_A$ and $D_{AB}$, are new for $n > 3$, whereas presemifields in family $D_B$ are isotopic to Generalized Twisted Fields for each $n\geq  3$.