Commutative Presemifields and Semifields

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Introduction

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 [math]\displaystyle{ \mathbb{S}=(\mathbb{F}_{p^n},+,\star) }[/math], for [math]\displaystyle{ p }[/math] a prime, [math]\displaystyle{ n }[/math] a positive integer, [math]\displaystyle{ \mathbb{S}=(\mathbb{F}_{p^n},+) }[/math] additive group and [math]\displaystyle{ x\star y }[/math] multiplication linear in each variable.

Two presemifields [math]\displaystyle{ \mathbb{S}_1=(\mathbb{F}_{p^n},+,\star) }[/math] and [math]\displaystyle{ \mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ) }[/math] are called isotopic if there exist three linear permutation [math]\displaystyle{ T,M,N }[/math] of [math]\displaystyle{ \mathbb{F}_{p^n} }[/math] such that [math]\displaystyle{ T(x\star y)=M(x)\circ N(y) }[/math], for any [math]\displaystyle{ x,y\in\mathbb{F}_{p^n} }[/math]. If [math]\displaystyle{ M=N }[/math] then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:

  • given [math]\displaystyle{ \mathbb{S}=(\mathbb{F}_{p^n},+,\star) }[/math] let [math]\displaystyle{ F_\mathcal{S}(x)=\frac{1}{2}(x\star x) }[/math];
  • given [math]\displaystyle{ F }[/math] let [math]\displaystyle{ \mathbb{S}_F=(\mathbb{F}_{p^n},+,\star) }[/math] defined by [math]\displaystyle{ x\star y=F(x+y)-F(x)-F(y) }[/math].

Hence two quadratic planar functions [math]\displaystyle{ F,F' }[/math] are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:

  • [math]\displaystyle{ F,F' }[/math] are CCZ-equivalent if and only if [math]\displaystyle{ \mathbb{S}_F,\mathbb{S}_{F'} }[/math] are strongly isotopic;
  • for [math]\displaystyle{ n }[/math] odd, isotopic coincides with strongly isotopic;
  • if [math]\displaystyle{ F,F' }[/math] are isotopic equivalent, then there exists a linear map [math]\displaystyle{ L }[/math] such that [math]\displaystyle{ F' }[/math] is EA-equivalent to [math]\displaystyle{ F(x+L(x))-F(x)-F(L(x)) }[/math].