The periodicity is proved for the arithmetic function defined as the quotient of the product of
k
+
1
k+1
values (where
k
≥
1
k \geq 1
) of a polynomial
f
∈
Z
[
x
]
f\in {\mathbb Z}[x]
at
k
+
1
k + 1
consecutive integers
f
(
n
)
f
(
n
+
1
)
⋯
f
(
n
+
k
)
{f(n) f(n + 1) \cdots f(n + k)}
and the least common multiple of the corresponding integers
f
(
n
)
f(n)
,
f
(
n
+
1
)
f(n + 1)
, …,
f
(
n
+
k
)
f(n + k)
. It is shown that this function is periodic if and only if no difference between two roots of
f
f
is a positive integer smaller than or equal to
k
k
. This implies an asymptotic formula for the least common multiple of
f
(
n
)
f(n)
,
f
(
n
+
1
)
f(n+1)
, …,
f
(
n
+
k
)
f(n+k)
and extends some earlier results in this area from linear and quadratic polynomials
f
f
to polynomials of arbitrary degree
d
d
. A period in terms of the reduced resultants of
f
(
x
)
f(x)
and
f
(
x
+
ℓ
)
f(x+\ell )
, where
1
≤
ℓ
≤
k
1 \leq \ell \leq k
, is given explicitly, as well as few examples of
f
f
when the smallest period can be established.