On the Automorphism Group of a Semifield of Order 54

The purpose of this article is to determine Aut(A) where A is a semifield of order 5⁴ admitting an automorphism group E ? Z ₂ × Z ₂ acting freely on A.     

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3,q), small Rédei minimal blocking sets of PG(2,q²), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface H(3,q²).     

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely, any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski-Ostrom polynomials over any finite field of order p³, p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X¹⁰+aX⁶−a²X² with a≠0 describes precisely two new commutative presemifields of order e³ for each odd e?5.     

The Autotopism Group of A Semifield of Order p 4, p is an Odd Prime

The aim of this article is to investigate the autotopism group of a semifield of order p ⁴, p is an odd prime, admitting a four-group of automorphisms E? Z ₂ × Z ₂ acting freely on A.     

Sur la Classification des Nilalgèbres Commutatives de Nilindice 3

We give some structure results and recursive-like methods for constructions and classifications of commutative nilalgebras of nilindex 3.     

Simplectic spreads and finite semifields

It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus.      

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract

Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A _t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A _t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A ₃ to the sedenions A ₄, revealing the symmetry lost on that stage.     

The BEL-rank of finite semifields

In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields.     

Commutative semifields from projection mappings

We describe a general projection method to construct commutative semifields in odd characteristic. One application yields a family of commutative semifields of order q ^2m with middle nucleus of order at least q ² for every odd prime-power q and every odd integer m > 1. Another application of the method yields a generalization of the Budaghyan–Helleseth family and also greatly simplifies the construction.     

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).     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Infinite families of new semifields

We construct six new infinite families of finite semifields, all of which are two-dimensional over their left nuclei. We give constructions for both even and odd characteristics when the left nucleus has odd dimension over the center. The characteristic is odd in the one family in which the left nucleus has even dimension over the center. Spread sets of linear maps are used in all the constructions.     

Semiprimality and Nilpotency of Nonassociative Rings Satisfying x(yz) = y(zx)

In this article we study nonassociative rings satisfying the polynomial identity x(yz) = y(zx), which we call “cyclic rings.” We prove that every semiprime cyclic ring is associative and commutative and that every cyclic right-nilring is solvable. Moreover, we find sufficient conditions for the nilpotency of cyclic right-nilrings and apply these results to obtain sufficient conditions for the nilpotency of cyclic right-nilalgebras.