Commutative presemifields and semifields

Abstract. Strong conditions are derived for when two commutative presemi-elds are isotopic. It is then shown that any commutative presemield of odd order can be described by a planar Dembowski-Ostrom polynomial and con-versely, any planar Dembowski-Ostrom polynomial describes a commutative presemi eld of odd order. These results allow a classication of all planar functions which describe presemields isotopic to a nite eld and of all pla-nar functions which describe presemields isotopic to Albert's commutative twisted elds. A classication of all planar Dembowski-Ostrom polynomials over any nite eld 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 pla-nar polynomials X10 + aX6 a2X2 with a 6 = 0 describes precisely two new commutative presemields of order 3e for each odd e 5. 1