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F p n .Then S =( F p n , + ,
for all x, y
) is a commutative semifield isotopic
to S with identity aa .Wesay S is a commutative semifield corresponding
to the commutative presemifield S . An isotopism between S and S is a strong
isotopism L a ( x ) ,L a ( x ) ,x with a linear permutation L a ( x )= ax ,see[9].
Let F be a planar DO polynomial over F p n .Then S =( F p n , + , ), with
xy = F ( x + y )
F ( x )
F ( y )
for any x, y
)
the commutative semifield corresponding to the commutative presemifield S with
F p n , is a commutative presemifield. We denote by S F =( F p n , + ,
isotopism L 1 ( x ) ,L 1 ( x ) ,x and we call S F =( F p n , + ,
)the commutative semi-
field defined by the planar DO polynomial F . Conversely, given a commutative
presemifield S =( F p n , + , ) of odd order, the function given by
F ( x )= 1
2 ( xx )
is a planar DO polynomial [9]. We prove in Section 4 that for planar DO polyno-
mials CCZ-equivalence coincides with linear equivalence. This implies that two
planar DO polynomials F and F are CCZ-equivalent if and only if the corre-
sponding commutative semifields S F and S F are strongly isotopic. It is proven
in [9] that for the n odd case two commutative presemifields are isotopic if and
only if they are strongly isotopic. There are also some sucient conditions for
the n even case when isotopy of presemifields implies their strong isotopy [9].
Thus, in the case n even it is potentially possible that isotopic commutative pre-
semifields define CCZ-inequivalent planar DO polynomials. However, in practice
no such cases are known.
Although commutative semifields have been intensively studied for more than
a hundred years, there are only eight distinct cases of known commutative semi-
fields of odd order (see [9]), and only three of them are defined for any odd
prime p . The eight distinct cases of known planar DO polynomials and corre-
sponding commutative semifields are the following:
x 2
(i)
over F p n which corresponds to the finite field F p n ;
(ii) x p t +1
over F p n ,with n/ gcd( t, n ) odd, which correspond to Albert's commutative
twisted fields [1,11,16];
(iii) the functions over F p 2 k , which correspond to the Dickson semifields [12];
(iv)
x 2
over F 3 n ,with n odd, corresponding to the Coulter-Matthews and Ding-
Yuan semifields [8,14];
(v) the function over F 3 2 k ,with k odd, corresponding to the Ganley semifield
[15];
(vi) the function over F 3 2 k corresponding to the Cohen-Ganley semifield [7];
x 10
x 6
±
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