Commutative Presemifields and Semifields

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A presemifield is a ring with left and right distributivity and with no zero divisor. A presemifield with a multiplicative identity is called a semifield. Any finite presemifield can be represented by , for a prime, a positive integer, additive group and multiplication linear in each variable.

Two presemifields and are called isotopic if there exist three linear permutations of such that , for any . If then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

  • given let ;
  • given let defined by .

Hence two quadratic planar functions are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

  • are CCZ-equivalent if and only if are strongly isotopic;
  • for odd, isotopic coincides with strongly isotopic;
  • if are isotopic equivalent, then there exists a linear map such that is EA-equivalent to .