AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$
F
(
R
)
of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$
F
(
R
)
×
is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$
R
[
x
]
/
(
x
2
)
=
R
[
α
]
, the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$
P
R
(
R
[
α
]
)
, consisting of those polynomial permutations of $$R[\alpha ]$$
R
[
α
]
represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$
F
(
R
)
×
by the group $${\mathcal{P}}(R)$$
P
(
R
)
of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$
R
=
F
q
, we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$
P
F
q
(
F
q
[
α
]
)
≅
P
(
F
q
)
⋉
θ
F
(
F
q
)
×
. Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$
p
n
and obtain canonical representations for these functions.
On the positivity of symmetric polynomial functions. Part III: Extremal polynomials of degree 4
