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 p(3), p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X-10 + aX(6) - a(2)X(2) with a not equal 0 describes precisely two new commutative presemifields of order 3(e) for each odd e >= 5. (c) 2007 Elsevier Inc. All rights reserved.