AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$
α
∈
[
0
,
1
]
, let $${\mathcal {B}}(n, \alpha )$$
B
(
n
,
α
)
denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$
A
⊆
{
1
,
…
,
n
}
is constructed by picking independently each element of $$\{1, \ldots , n\}$$
{
1
,
…
,
n
}
with probability $$\alpha $$
α
. Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$
A
.Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$
[
k
]
q
:
=
1
+
q
+
q
2
+
⋯
+
q
k
-
1
∈
Z
[
q
]
be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$
X
:
=
deg
lcm
(
[
A
]
q
)
, where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$
[
A
]
q
:
=
{
[
k
]
q
:
k
∈
A
}
. Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.
Polynomials Consisting of Quadratic Factors with Roots Modulo Any Positive Integer
