Let
R
\mathrm {R}
be a real closed field and let
Q
{\mathcal Q}
and
P
{\mathcal P}
be finite subsets of
R
[
X
1
,
…
,
X
k
]
\mathrm {R}[X_1,\ldots ,X_k]
such that the set
P
{\mathcal P}
has
s
s
elements, the algebraic set
Z
Z
defined by
⋀
Q
∈
Q
Q
=
0
\bigwedge _{Q \in {\mathcal Q}}Q=0
has dimension
k
′
k’
and the elements of
Q
{\mathcal Q}
and
P
{\mathcal P}
have degree at most
d
d
. For each
0
≤
i
≤
k
′
,
0 \leq i \leq k’,
we denote the sum of the
i
i
-th Betti numbers over the realizations of all sign conditions of
P
{\mathcal P}
on
Z
Z
by
b
i
(
P
,
Q
)
b_i({\mathcal P},{\mathcal Q})
. We prove that
\[
b
i
(
P
,
Q
)
≤
∑
j
=
0
k
′
−
i
(
s
j
)
4
j
d
(
2
d
−
1
)
k
−
1
.
b_i({\mathcal P},{\mathcal Q}) \le \sum _{j=0}^{k’ - i} {s \choose j} 4^{j} d(2d-1)^{k-1}.
\]
This generalizes to all the higher Betti numbers the bound
(
s
k
′
)
O
(
d
)
k
{s \choose k’}O(d)^k
on
b
0
(
P
,
Q
)
b_0({\mathcal P},{\mathcal Q})
. We also prove, using similar methods, that the sum of the Betti numbers of the intersection of
Z
Z
with a closed semi-algebraic set, defined by a quantifier-free Boolean formula without negations with atoms of the form
P
≥
0
P \geq 0
or
P
≤
0
P\leq 0
for
P
∈
P
P\in {\mathcal P}
, is bounded by
\[
∑
i
=
0
k
′
∑
j
=
0
k
′
−
i
(
s
j
)
6
j
d
(
2
d
−
1
)
k
−
1
,
\sum _{i = 0}^{k’}\sum _{j = 0}^{k’ - i} {s \choose j} 6^{j} d(2d-1)^{k-1},
\]
making the bound
s
k
′
O
(
d
)
k
s^{k’} O(d)^k
more precise.