F-q-pseudoreguli of PG(3, q(3)) and scattered semifields of order q(6)

In this paper, we study rank two semifields of order $q^6$ that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in \cite{MaPoTrSub} for $q\equiv 1(mod\,3)$. Here, we construct new infinite families of rank two scattered semifields for any $q$ odd prime power, with $q \equiv 1 (mod\,3)$; for any $q=2^{2h}$, such that $h \equiv 1 (mod\,3)$ and for any $q=3^h$ with $h \equiv \hspace{-0.3cm}/ \,\,0\, (mod\,3).$ Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields.