Difference between revisions of "Commutative Presemifields and Semifields"

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=Introduction=
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A <span class="definition">presemifield</span> is a ring with left and right distributivity and with no zero divisor.
 
A <span class="definition">presemifield</span> is a ring with left and right distributivity and with no zero divisor.
 
A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>.
 
A presemifield with a multiplicative identity is called a <span class="definition">semifield</span>.
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for <math>p</math> a prime, <math>n</math> a positive integer, <math>\mathbb{S}=(\mathbb{F}_{p^n},+)</math> additive group and <math>x\star y</math> multiplication linear in each variable.
 
for <math>p</math> a prime, <math>n</math> a positive integer, <math>\mathbb{S}=(\mathbb{F}_{p^n},+)</math> additive group and <math>x\star y</math> multiplication linear in each variable.
  
Two presemifields <math>\mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)</math> and <math>\mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)</math> are called <span class="definition">isotopic</span>  if there exist three linear permutation <math>T,M,N</math> of <math>\mathbb{F}_{p^n}</math> such that
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Two presemifields <math>\mathbb{S}_1=(\mathbb{F}_{p^n},+,\star)</math> and <math>\mathbb{S}_2=(\mathbb{F}_{p^n},+,\circ)</math> are called <span class="definition">isotopic</span>  if there exist three linear permutations <math>T,M,N</math> of <math>\mathbb{F}_{p^n}</math> such that
 
<math>T(x\star y)=M(x)\circ N(y)</math>,
 
<math>T(x\star y)=M(x)\circ N(y)</math>,
 
for any <math>x,y\in\mathbb{F}_{p^n}</math>. If <math>M=N</math> then they are called <span class="definition">strongly isotopic</span>.
 
for any <math>x,y\in\mathbb{F}_{p^n}</math>. If <math>M=N</math> then they are called <span class="definition">strongly isotopic</span>.
 
Each commutative presemifields of odd order defines a [[Planar Functions|planar]] DO polynomial and viceversa:
 
Each commutative presemifields of odd order defines a [[Planar Functions|planar]] DO polynomial and viceversa:
* given <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> let <math>F_\mathcal{S}(x)=\frac{1}{2}(x\star x)</math>;
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* given <math>\mathbb{S}=(\mathbb{F}_{p^n},+,\star)</math> let <math>F_\mathbb{S}(x)=\frac{1}{2}(x\star x)</math>;
 
* given <math>F</math> let <math>\mathbb{S}_F=(\mathbb{F}_{p^n},+,\star)</math> defined by <math>x\star y=F(x+y)-F(x)-F(y)</math>.
 
* given <math>F</math> let <math>\mathbb{S}_F=(\mathbb{F}_{p^n},+,\star)</math> defined by <math>x\star y=F(x+y)-F(x)-F(y)</math>.
  

Revision as of 11:29, 29 August 2019

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 .