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 = 0 describes precisely two new commutative presemifields of order 3 e for each odd e ≥ 5.

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.

We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 3 8 with left nucleus of order 3 and middle nucleus of order 3 2 .

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