Planar polynomials for commutative semifields with specified nuclei. (English) Zbl 1215.12012

Summary: 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 [Adv. Math. 217, No. 1, 282–304 (2008; Zbl 1194.12007)]. 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 \(3^2\).


12K10 Semifields
11T06 Polynomials over finite fields


Zbl 1194.12007


Full Text: DOI


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