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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
±
−