Let
N
N
be the number of solutions
(
u
1
,
…
,
u
n
)
(u_1,\ldots ,u_n)
of the equation
a
1
u
1
d
1
+
⋯
+
a
n
u
n
d
n
=
0
a_1u_1^{d_1}+\cdots +a_nu_n^{d_n}=0
over the finite field
F
q
F_q
, and let
I
I
be the number of solutions of the equation
∑
i
=
1
n
x
i
/
d
i
≡
0
(
mod
1
)
,
1
⩽
x
i
⩽
d
i
−
1
\sum _{i=1}^nx_i/d_i\equiv 0\pmod {1}, 1\leqslant x_i\leqslant d_i-1
. If
I
>
0
I>0
, let
L
L
be the least integer represented by
∑
i
=
1
n
x
i
/
d
i
,
1
⩽
x
i
⩽
d
i
−
1
\sum _{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1
.
I
I
and
L
L
play important roles in estimating
N
N
. Based on a partition of
{
d
1
,
…
,
d
n
}
\{d_1,\dots ,d_n\}
, we obtain the factorizations of
I
,
L
I, L
and
N
N
, respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for
N
N
in some special cases.