We determine a lower bound for the dimension of the Čech cohomology of the root sets of maps from the sphere
S
2
n
+
1
S^{2n+1}
and from the real projective space
R
P
2
n
+
1
{\mathbb {R}\mathrm {P}}^{2n+1}
into the complex projective space
C
P
n
{\mathbb {C}\mathrm {P}}^n
, for
n
≥
1
n\geq 1
. For each such a map, we construct a representative of its homotopy class which realize the lower bound and whose root set is minimal in the class. We prove that the circle is a minimal root set for any non-trivial homotopy class. We present analogous results for maps from both
S
4
n
+
3
S^{4n+3}
and
R
P
4
n
+
3
{\mathbb {R}\mathrm {P}}^{4n+3}
into the orbit space
C
P
2
n
+
1
/
τ
{\mathbb {C}\mathrm {P}}^{2n+1}\!/\tau
, for
n
≥
0
n\geq 0
, where
τ
\tau
is a free involution on
C
P
2
n
+
1
{\mathbb {C}\mathrm {P}}^{2n+1}
. In this setting, we prove that the disjoint union of two circles is a minimal root set for any non-trivial homotopy class.