# Difference between revisions of "Commutative Presemifields and Semifields"

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 ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$, for ${\displaystyle p}$ a prime, ${\displaystyle n}$ a positive integer, ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+)}$ additive group and ${\displaystyle x\star y}$ multiplication linear in each variable.
Two presemifields ${\displaystyle \mathbb {S} _{1}=(\mathbb {F} _{p^{n}},+,\star )}$ and ${\displaystyle \mathbb {S} _{2}=(\mathbb {F} _{p^{n}},+,\circ )}$ are called isotopic if there exist three linear permutations ${\displaystyle T,M,N}$ of ${\displaystyle \mathbb {F} _{p^{n}}}$ such that ${\displaystyle T(x\star y)=M(x)\circ N(y)}$, for any ${\displaystyle x,y\in \mathbb {F} _{p^{n}}}$. If ${\displaystyle M=N}$ then they are called strongly isotopic. Each commutative presemifields of odd order defines a planar DO polynomial and viceversa:
• given ${\displaystyle \mathbb {S} =(\mathbb {F} _{p^{n}},+,\star )}$ let ${\displaystyle F_{\mathbb {S} }(x)={\frac {1}{2}}(x\star x)}$;
• given ${\displaystyle F}$ let ${\displaystyle \mathbb {S} _{F}=(\mathbb {F} _{p^{n}},+,\star )}$ defined by ${\displaystyle x\star y=F(x+y)-F(x)-F(y)}$.
Hence two quadratic planar functions ${\displaystyle F,F'}$ are isotopic equivalent if their corresponding presemifields are isotopic. Moreover, we have:
• ${\displaystyle F,F'}$ are CCZ-equivalent if and only if ${\displaystyle \mathbb {S} _{F},\mathbb {S} _{F'}}$ are strongly isotopic;
• for ${\displaystyle n}$ odd, isotopic coincides with strongly isotopic;
• if ${\displaystyle F,F'}$ are isotopic equivalent, then there exists a linear map ${\displaystyle L}$ such that ${\displaystyle F'}$ is EA-equivalent to ${\displaystyle F(x+L(x))-F(x)-F(L(x))}$.