A continuous map
f
:
X
→
R
N
f:X\to \mathbb {R} ^{N}
is said to be
k
k
-regular if whenever
x
1
,
…
,
x
k
x_{1},\dots , x_{k}
are distinct points of
X
X
, then
f
(
x
1
)
,
…
,
f
(
x
k
)
f(x_{1}),\dots , f(x_{k})
are linearly independent over
R
\mathbb {R}
. For smooth manifolds
M
M
we obtain new lower bounds on the minimum
N
N
for which a
2
k
2k
-regular map
M
→
R
N
M \to \mathbb {R} ^{N}
can exist in terms of the dual Stiefel-Whitney classes of
M
M
.