We consider the KZ differential equations over
C
\mathbb {C}
in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field
F
p
\mathbb {F}_p
. We study the polynomial solutions of these differential equations over
F
p
\mathbb {F}_p
, constructed in a previous work joint with V. Schechtman and called the
F
p
\mathbb {F}_p
-hypergeometric solutions.
The dimension of the space of
F
p
\mathbb {F}_p
-hypergeometric solutions depends on the prime number
p
p
. We say that the KZ equations have ample reduction for a prime
p
p
, if the dimension of the space of
F
p
\mathbb {F}_p
-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over
C
\mathbb {C}
. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis
F
p
\mathbb {F}_p
-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials
(
z
i
−
z
j
)
M
i
+
M
j
(z_i-z_j)^{M_i+M_j}
are replaced with
(
z
i
−
z
j
)
M
i
+
M
j
−
p
(z_i-z_j)^{M_i+M_j-p}
and the Euler gamma function
Γ
(
x
)
\Gamma (x)
is replaced with a suitable
F
p
\mathbb {F}_p
-analog
Γ
F
p
(
x
)
\Gamma _{\mathbb {F}_p}(x)
defined on
F
p
\mathbb {F}_p
.