Let
F
1
F_1
, …,
F
R
F_R
be homogeneous polynomials of degree
d
⩾
2
d\geqslant 2
with integer coefficients in
n
n
variables, and let
F
=
(
F
1
,
…
,
F
R
)
\mathbf {F}=(F_1,\ldots ,F_R)
. Suppose that
F
1
F_1
, …,
F
R
F_R
is a non-singular system and
n
⩾
4
d
+
2
d
2
R
5
n\geqslant 4^{d+2}d^2R^5
. We prove that there are infinitely many solutions to
F
(
x
)
=
0
\mathbf {F}(\mathbf {x})=\mathbf {0}
in prime coordinates if (i)
F
(
x
)
=
0
\mathbf {F}(\mathbf {x})=\mathbf {0}
has a non-singular solution over the
p
p
-adic units
U
p
\mathbb {U}_p
for all prime numbers
p
p
, and (ii)
F
(
x
)
=
0
\mathbf {F}(\mathbf {x})=\mathbf {0}
has a non-singular solution in the open cube
(
0
,
1
)
n
(0,1)^n
.