Notation 4.1. For a finite set of polynomials $\mathcal {Q} \subset k[X_1,\ldots ,X_N]$ , and an algebraic set $W \subset k^N$ , we denote by $\mathrm {Z}(\mathcal {Q},W)$ the set of common zeros of $\mathcal {Q}$ in V. If $\mathcal {Q} = \{Q\}$ , we write $\mathrm {Z}(Q,W)$ for $\mathrm {Z}(\mathcal {Q},W)$ .

Proof. Let ${\mathcal P} = \{P_1,\ldots ,P_s\}$ (the implicit ordering of the polynomials in ${\mathcal P}$ will play a role in the proof).

First consider a sequence of p polynomials of ${\mathcal P}$ , $P_{i_1},\ldots ,P_{i_p},$ with $1 \leq i_1 < \cdots < i_p \leq s$ . Let $\{W_{\alpha _1}\}_{\alpha _1 = 1,2..}$ denote the irreducible components of $\mathrm {Z}(P_{i_1},V)$ . Similarly, let $\{W_{\alpha _1,\alpha _2}\}_{\alpha _2 = 1,2,\ldots }$ denote the irreducible components of $W_{\alpha _1} \cap \mathrm {Z}(P_{i_2},V)$ and so on. We prove by induction that for $1 \leq \ell \leq p$

(4.1) $$ \begin{align} \sum_{\alpha_1,\alpha_2,\ldots,\alpha_\ell} \deg(W_{\alpha_1,\ldots,\alpha_\ell}) \leq D \cdot d^\ell. \end{align} $$

The claim is trivially true for $\ell =1$ . Suppose it is true up to $\ell -1 $ . Then, using Theorem H, we get that

$$\begin{align*}\sum_{\alpha_1,\ldots,\alpha_{\ell-1}} \deg(\mathrm{Z}(P_\ell,W_{\alpha_1,\ldots,\alpha_{\ell-1}})) \leq D \cdot d^{\ell-1}\cdot d = D \cdot d^\ell. \end{align*}$$

Now suppose that $\dim V_{\rho ,i} = 0$ and let $V_{\rho ,i} = \{x\}$ .

We first claim that there must exist $P_{i_1},\ldots ,P_{i_p} \in {\mathcal P}_{\rho }$ , with $1 \leq i_1 < i_2 < \cdots < i_p \leq s$ , and irreducible affine algebraic varieties $V_0, V_1,\ldots ,V_p$ satisfying:

1. A. $V= V_0 \supset V_1 \supset V_2 \supset \cdots \supset V_p = \{x\}$ , and

2. B. for each $j, \;1 \leq j \leq p$ , $V_{j}$ is an irreducible component of $\mathrm {Z}(P_{i_j},V_{j-1})$ which contains x.

Note that in the sequence, $V_0,V_1,\ldots ,V_p$ , $\dim (V_i)$ is necessarily equal to $p-i$ . To prove the claim, let

$$\begin{align*}{\mathcal P}_\rho = \{P_{j_1},\ldots,P_{j_m}\}, \end{align*}$$

with $1 \leq j_1 < j_2 < \cdots < j_m \leq s$ . Let $W_0 = V$ and for each $h, 1 \leq h \leq m$ , define inductively $W_{h}$ to be an irreducible component of $\mathrm {Z}(P_{j_h},W_{h-1})$ containing the point x. By definition, $V = W_0 \supset W_1 \supset \cdots \supset W_m = \{x\}$ and each $W_h$ is irreducible. Hence, there must exist precisely p indices, $1 \leq \alpha _1 < \cdots < \alpha _p \leq m$ such that $\dim W_{\alpha _{i}} = \dim W_{\alpha _{i-1}} - 1$ . If $\beta \not \in \{\alpha _1,\ldots ,\alpha _p\}$ , then $W_{\beta } = W_{\beta - 1}$ . Now, let $i_1 = j_{\alpha _1},\ldots , i_p = j_{\alpha _p}$ , and $V_j = W_{\alpha _j}$ for $0 \leq j \leq p$ . It is clear from construction that the sequence $V_0,\ldots ,V_p$ satisfies the properties stated above, thus proving the claim.

The number of sequences of polynomials of length p, $P_{i_1},\ldots ,P_{i_p},$ with $1 \leq i_1 < \cdots < i_p \leq s$ , is $\binom {s}{p}$ . The lemma now follows from the bound in (4.1).

Proof. In order to bound the degree of $V_{\rho ,i}$ with $\dim V_{\rho ,i} =\ell $ , it suffices to count the isolated zeros of the intersection of $V_{\rho ,i}$ with a generic affine subspace L of dimension $p-\ell $ . Since there are only a finite number of $V_{\rho ,i}$ ’s to consider, we can assume that

1. A. L is generic for all of them,

2. B. the sets $V_{\rho ,i} \cap L$ are pairwise disjoint,

3. C. for each $V_{\rho ,i}$ with $\dim _{\mathrm {C}}V_{\rho ,i} =\ell $ , $ \mathrm {Z}(P,L \cap V_{\rho ,i}) = \emptyset $ for each $P \in {\mathcal P} \setminus {\mathcal P}_\rho $ .

Now, restrict to the subspace L and apply Lemma 4.2, noting that an isolated point of $V_{\rho ,i} \cap L$ is an isolated point of the family ${\mathcal P}$ restricted to L.

Proof of Theorem 2.1.

Follows directly from Lemma 4.3.

Notation 4.5. For $\mathrm {R}$ a real closed field, we denote by $\mathrm {R} \left \langle {\varepsilon } \right \rangle $ the real closed field of algebraic Puiseux series in ${\varepsilon }$ with coefficients in $\mathrm {R}$ . We use the notation $\mathrm {R} \left \langle {\varepsilon }_{1}, \ldots , {\varepsilon }_{m} \right \rangle $ to denote the real closed field $\mathrm {R} \left \langle {\varepsilon }_{1} \right \rangle \left \langle {\varepsilon }_{2} \right \rangle \cdots \left \langle {\varepsilon }_{m} \right \rangle $ . Note that in the unique ordering of the field $\mathrm {R} \left \langle {\varepsilon }_{1}, \ldots , {\varepsilon }_{m} \right \rangle $ , $0< {\varepsilon }_{m} \ll {\varepsilon }_{m-1} \ll \cdots \ll {\varepsilon }_{1} \ll 1$ .

Notation 4.6. For elements $x \in \mathrm {R} \left \langle {\varepsilon } \right \rangle $ which are bounded over $\mathrm {R}$ , we denote by $\lim _{{\varepsilon }} x$ the image in $\mathrm {R}$ under the usual map that sets ${\varepsilon }$ to $0$ in the Puiseux series x.

Notation 4.7. If $\mathrm {R}'$ is a real closed extension of a real closed field $\mathrm {R}$ , and $S \subset \mathrm {R}^{k}$ is a semi-algebraic set defined by a first-order formula with coefficients in $\mathrm {R}$ , then we will denote by $\mathrm {Ext}(S, \mathrm {R}') \subset \mathrm {R}^{\prime {k}}$ the semi-algebraic subset of $\mathrm {R}^{\prime {k}}$ defined by the same formula. It is well known that $\mathrm {Ext}(S, \mathrm {R}')$ does not depend on the choice of the formula defining S [Reference Basu, Pollack and Roy5, Proposition 2.87].

Proof. Let $x \in C$ . Then, $x \in {\mathcal R}(\widetilde {\phi }_\sigma ,\mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}))$ , and let $\widetilde {C}_\sigma $ be the semi-algebraically connected component of ${\mathcal R}(\widetilde {\phi }_\sigma ,\mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}))$ containing x. Then $\widetilde {C}_\sigma \cap V(\mathrm {R}) = C \cap V(\mathrm {R})$ (see [Reference Basu, Pollack and Roy5, Proposition 13.7]).

Proof. Follows from the fact that $\alpha (i)\gamma _i, i \in I$ are independent transcendentals over $\mathrm {C}$ .

Proof. Let $\mathcal {Q}'$ be a maximal subset of $\mathcal {Q}$ with the property that $\mathrm {Z}(\mathcal {Q}',\mathrm {R}^N) \cap C \neq \emptyset $ . Then, $\mathcal {Q}'$ satisfies the required property in the lemma (see [Reference Basu, Pollack and Roy5, Proposition 13.1]).

Proof. Follows from Lemmas 4.11 and 4.14.

Proof. Let $\widetilde {S}$ be the intersection of $\mathrm {Ext}(S,\mathrm {R}{\langle }\zeta {\rangle })$ with the set defined by $-\zeta \leq Q \leq \zeta $ . Let $C \in \mathrm {Cc}(\mathrm {Z}(Q,S))$ and $x \in C$ . Then, it follows from the hypothesis that $\dim _x \mathrm {Z}(Q,C) < \dim _x S$ . Hence, for all $r>0$ , denoting (here and elsewhere) by $B_N(x,r)$ the open Euclidean ball centered at x of radius r, $\mathrm {Z}(Q,C) \cap B_N(x,r)$ is nonempty and properly contained in $S \cap B_N(x,r)$ . So $S - \mathrm {Z}(Q,C) \neq \emptyset $ and $x \in \overline {S - \mathrm {Z}(Q,C)}$ . Applying the semi-algebraic curve selection lemma (see, e.g., [Reference Basu, Pollack and Roy5, Theorem 3.19]), there exists a semi-algebraic curve $\gamma :[0,1] \rightarrow S$ , such that $\gamma (0) = x$ , and $\gamma ((0,1]) \subset S - \mathrm {Z}(Q,C)$ . In particular, $Q(\gamma (1)) \neq 0$ .

Suppose without loss of generality that $Q(\gamma (1))> 0$ . Using the intermediate value property of the field $\mathrm {R}{\langle }\zeta {\rangle }$ ([Reference Basu, Pollack and Roy5, Proposition 4.3]), there exists $\widetilde {t} \in \mathrm {R}{\langle }\zeta {\rangle }, \widetilde {t}> 0$ , such that $Q(\mathrm {Ext}(\gamma ,\mathrm {R}{\langle }\zeta {\rangle })(\widetilde {t})) = \zeta $ , and let $\widetilde {x} = \mathrm {Ext}(\gamma ,\mathrm {R}{\langle }\zeta {\rangle })(\widetilde {t})$ . Then, $\widetilde {x} \in \mathrm {Z}(Q-\zeta , \mathrm {Ext}(S,\mathrm {R}{\langle }\zeta {\rangle }))$ . Let $\widetilde {C} \in \mathrm {Cc}(\mathrm { Z}(Q-\zeta , \mathrm {Ext}(S,\mathrm {R}{\langle }\zeta {\rangle })))$ with $\widetilde {x} \in \widetilde {C}$ . Then, $\lim _\zeta \widetilde {C} \subset C$ . Thus we have proved that for each $C \in \mathrm {Cc}(\mathrm {Z}(Q,S))$ , there exists $\widetilde {C} \in \mathrm {Cc}(\mathrm {Z}(Q-\zeta , \mathrm {Ext}(S,\mathrm {R}{\langle }\zeta {\rangle }))) \cup \mathrm {Cc}(\mathrm { Z}(Q+\zeta , \mathrm {Ext}(S,\mathrm {R}{\langle }\zeta {\rangle })))$ , with $\lim _\zeta \widetilde {C} \subset C$ . The lemma follows.

Proof of Proposition 4.8.

Using Lemma 4.15, we are reduced to bounding the number of semi-algebraically connected components of the real algebraic sets,

$$\begin{align*}\bigcap_{i\in I} \mathrm{Z}(P_i + \alpha(i)\gamma_i,\mathrm{Ext}(V(\mathrm{R}),\widetilde{\mathrm{R}})), \end{align*}$$

and

$$\begin{align*}\mathrm{Z}(P_0 - {\varepsilon} \gamma_0,\mathrm{Ext}(V(\mathrm{R}),\widetilde{\mathrm{R}})) \cap \bigcap_{i\in I} \mathrm{Z}(P_i + \alpha(i)\gamma_i,\mathrm{Ext}(V(\mathrm{R}),\widetilde{\mathrm{R}})), \end{align*}$$

where $I \subset [1,s]$ and $\alpha $ is as defined in Lemma 4.15. Using Lemma 4.12 and Remark 4.13, we only need to consider I with $\mathrm {card}(I) \leq p$ for the algebraic sets $\bigcap _{i\in I} \mathrm {Z}(P_i + \alpha (i)\gamma _i,\mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}))$ and of cardinality at most $p-1$ for the algebraic sets $\mathrm {Z}(P_0 - {\varepsilon } \gamma _0,\mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}})) \cap \bigcap _{i\in I} \mathrm {Z}(P_i + \alpha (i)\gamma _i,\mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}))$ .

Thus in order to bound $\sum _{\sigma \in \{0,1,-1\}^{\mathcal {P}}} b_0({\mathcal R}(\sigma ,V(\mathrm {R})))$ it suffices to bound (using Lemma 4.16 and Remark 4.17)

1. A. $b_0(\mathrm {Z}(Q_{I,\alpha } - \zeta , \mathrm {Ext}(V,\widetilde {\mathrm {R}}{\langle }\zeta {\rangle })))$ , $I \subset [1,s], 1 \leq \mathrm {card}(I) \leq p$ ,

2. B. $b_0(\mathrm {Z}(Q_{I,\alpha ,0} - \zeta , \mathrm {Ext}(V,\widetilde {\mathrm {R}}{\langle }\zeta {\rangle })))$ , $I \subset [1,s], 1 \leq \mathrm {card}(I) < p$ , and

3. C. $b_0(V(\mathrm {R}))$ .

Item (A) corresponds to A, item (B) to B, and item (C) to C in the right-hand side of inequality (4.5).

Proof. It follows from the Noether normalization theorem (see, e.g., [Reference de Jong and Pfister13, Theorem 2.2.9]) that for a generic linear projection map $P:\mathrm {C}^{N} \rightarrow \mathrm {C}^p$ of rank p, $V_\ell := P^{-1}(\ell ) \cap V$ is (nonempty and) finite for all $\ell \in L = \mathrm {Im}(P) \subset \mathrm {C}^N$ .

Let $P_0: \mathrm {C}^N \rightarrow \mathrm {C}^N$ be a generic projection of rank p, and denote $L_0 = \mathrm {Im}(P_0) \subset \mathrm {C}^N$ .

The set $\mathcal {Q}$ of linear projections $Q:\mathrm {C}^N \rightarrow \mathrm {C}^N$ of rank $p+1$ such that $L_0 \subset \mathrm {Im}(Q) $ is a subvariety of the variety of all projections of rank $p+1$ .

Let

$$\begin{align*}\Gamma = \{\ell,Q) \in L_0 \times \mathcal{Q} \mid Q|_{V_\ell} \text{ is not injective}\}. \end{align*}$$

Notice that $\dim \Gamma < p + \dim \mathcal {Q}$ .

Let $\pi _1: \Gamma \rightarrow U$ , $\pi _2: \Gamma \rightarrow \mathcal {Q}$ be the two projection maps.

For a generic choice of $Q_0 \in \mathcal {Q}$ ,

(4.6) $$ \begin{align} \dim \pi_2^{-1}(Q_0) < p, \end{align} $$

since otherwise $\dim \Gamma = p + \dim \mathcal {Q}$ .

Let $W' = Q_0(P_0^{-1}(\pi _1(\pi _2^{-1}(Q_0))))$ . Then $\dim W' < p$ (using equation (4.6) and the fact that $P_0|_V$ is a finite map).

Now define $\pi : \mathrm {C}^{N} \rightarrow \mathrm {Im}(Q_0) \cong \mathrm {C}^{p+1}$ , by $\pi (x) = Q_0(x)$ . Then, with $W = \pi (V)$ , $W' \subset W$ , with $\dim (W') < p = \dim (W)$ . Hence, we have for each $w \in W - W'$ , $\mathrm {card}(\pi ^{-1}(w) \cap V) = 1$ . This proves the lemma.

Proof. It follows from the Noether normalization theorem (see, e.g., [Reference de Jong and Pfister13, Theorem 2.2.9]) that for $\pi $ generic, $\pi (V)$ is a hypersurface in $\mathrm {C}^{p+1}$ .

Let $w \in W = \pi (V)$ , and let $V_w = \pi ^{-1}(w) \cap V$ . It follows from Lemma 4.18 that there exists a neighborhood $U_w \subset W$ of w in W, such that each $z \in V_w$ has a neighborhood $U_{z}$ in $\pi ^{-1}(U_w) \cap V$ , such that $\pi |_{U_z}$ is injective, and moreover for $z \neq z' \in V_w$ , the neighborhoods $U_{z}$ and $U_{z'}$ are disjoint, and moreover it follows from Lemma 4.18 that $\pi (U_z) \neq \pi (U_{z'})$ .

Let $\widetilde {\pi }: \widetilde {U} \rightarrow U_w$ (shrinking $U_w$ if necessary) be a blow-up of $U_w$ at w. By the universal property of the blow-up morphism (see, e.g., [Reference Hartshorne20, Chapter II, Proposition 7.14]) there exists a unique morphism $\pi ': \pi ^{-1}(U_w) \rightarrow \widetilde {U}$ , such that $\pi |_{\pi ^{-1}{U_w}} = \widetilde {\pi } \circ \pi '$ . It follows from the fact that $\pi (U_z) \neq \pi (U_{z'})$ , that $\pi '(z) \neq \pi '(z')$ . But $\pi '(z), \pi '(z')$ belong to the exceptional divisor of the blow-up $\widetilde {\pi }$ . It now follows from the definition of multiplicity (see Section 4.2.1) that $\mathrm {mult}_w(W)>1$ , and it follows from Lemma 4.4 that in this case $w \in \mathrm {Sing}(W)$ .

Proof. It follows from Lemma 4.19 that for any $z \in \pi _{\mathrm {C}}(V) - \mathrm {Sing}(\pi _{\mathrm {C}}(V))$ ,

$$\begin{align*}\mathrm{card}(\pi_{\mathrm{C}}^{-1}(z) \cap V) = 1. \end{align*}$$

Now, since V is defined over $\mathrm {R}$ , for $x \in \pi (V(\mathrm {R}))$ , $\mathrm {card}(\pi _{\mathrm {C}}^{-1}(x) \cap V) = 1$ implies that $\pi _{\mathrm {C}}^{-1}(x) \cap V \subset V(\mathrm {R})$ , since otherwise $\pi _{\mathrm {C}}^{-1}(x) \cap V$ would contain at least one pair of conjugate points $z \neq \bar {z}, z,\bar {z} \in \pi _{\mathrm {C}}^{-1}(x) \cap V$ .

Taken together, the previous two statements imply:

$$\begin{align*}\pi_{\mathrm{C}}^{-1}(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))) \cap V \subset V(\mathrm{R}). \end{align*}$$

We now prove that for each $C \in \mathrm {Cc}(V(\mathrm {R}))$ , there exists $D \in \mathrm {Cc}(\pi _{\mathrm {C}}(V)(\mathrm {R}) - \mathrm {Sing}(\pi _{\mathrm {C}}(V)))$ , such that $\pi ^{-1}(D) \subset C$ . It would follow that

$$\begin{align*}b_0(V(\mathrm{R})) \leq b_0(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))). \end{align*}$$

Let $C \in \mathrm {Cc}(V(\mathrm {R}))$ . Since $\pi (V(\mathrm {R})) \subset \pi _{\mathrm {C}}(V)(\mathrm {R})$ , there exists $D' \in \mathrm {Cc}(\pi _{\mathrm {C}}(V)(\mathrm {R}))$ such that $\pi (C) \subset D'$ .

Since the local real dimension of $V(\mathrm {R})$ is equal to p at all points of $V(\mathrm {R})$ , $\dim _{\mathrm {R}} C = \dim _{\mathrm {R}} D' = p$ . (This is the place where the equidimensionality property of V plays a role.) Moreover, $\dim _{\mathrm {R}} D' \cap \mathrm {Sing}(\pi _C(V)) < p$ , and hence $D' - \mathrm {Sing}(\pi _C(V)) \neq \emptyset $ . Now let

$$\begin{align*}D \in \mathrm{Cc}(D' - \mathrm{Sing}(\pi_C(V))) \subset \mathrm{Cc}(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))). \end{align*}$$

Then, $\pi ^{-1}(D) \subset C$ and we are done.

Proof. Let $Q = \sum _{i=1}^m \left (\frac {\partial P}{\partial X_i}\right )^2$ . Then, there exists $R> 0$ , such that

$$\begin{align*}b_0(V(\mathrm{R}) - \mathrm{Sing}(V)) = b_0((V(\mathrm{R}) - \mathrm{Sing}(V)) \cap B_m(0,R)), \end{align*}$$

and

$$\begin{align*}b_0((V(\mathrm{R}) - \mathrm{Sing}(V)) \cap B_m(0,R)) = b_0(S) \end{align*}$$

where $S \subset \mathrm {R}{\langle }{\varepsilon }{\rangle }^N$ is defined by

$$\begin{align*}(P = 0) \wedge (Q \geq {\varepsilon}) \wedge (\sum_{i=1}^m X_i^2 -R \leq 0). \end{align*}$$

It follows from Lemma 4.14 that $b_0(S)$ is at most the number of semi-algebraically connected components of the $\binom {3}{1} + \binom {3}{2} + \binom {3}{3} = 7$ algebraic sets defined by taking one, two, or three polynomials $P,Q - {\varepsilon }, \sum _{i=1}^m X_i^2 -R$ at a time.

The number of semi-algebraically connected components of each one of them is bounded by

$$\begin{align*}2d(4d -1)^{m-1}, \end{align*}$$

using the basic inequality on the Betti numbers of real algebraic sets due to Oleı̆nik and Petrovskiı̆ [Reference Petrovskiĭ and Oleĭnik31] (see, e.g., [Reference Basu, Pollack and Roy5, Proposition 7.28]). The lemma follows.

Proof of Proposition 4.9.

The proposition follows from Lemmas 4.18, 4.19, 4.20, and 4.21.

Proof of Proposition 4.10.

Note that for any $I \subset [1,s] \mathrm {Z}(Q_{I,\alpha ,0} - \zeta , \mathrm {Ext}(V,\widetilde {\mathrm {R}}{\langle }\zeta {\rangle }))$ and $\mathrm {Z}(Q_{I,\alpha } - \zeta , \mathrm {Ext}(V,\widetilde {\mathrm {R}}{\langle }\zeta {\rangle }))$ are real algebraic sets in $\widetilde {R}{\langle }\zeta {\rangle }[i]^N$ of dimension $p - 1$ if nonempty, satisfying the property of constancy of local real dimension, and of degree bounded by $2 D d$ .

The number of summands appearing in the definition of A (equation (4.2)) is equal to $\sum _{1 \leq j \leq p} 4^j \binom {s}{j}$ , and the number of summands appearing in the definition of B (equation (4.3)) is equal to $\sum _{1 \leq j \leq p-1}4^j \binom {s}{j}$ .

The proposition follows from applying Lemmas 4.20 and 4.21 to the algebraic sets $\mathrm {Z}(Q_{I,\alpha } - \zeta , \mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}{\langle }\zeta {\rangle }))$ and $\mathrm {Z}(Q_{I,\alpha ,0} - \zeta , \mathrm {Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}{\langle }\zeta {\rangle }))$ , and the combinatorial count on the number of terms appearing in the right-hand side of the definitions of A and B stated in the previous paragraph.

Proof of Theorem 2.5.

First of all, observe that using the fact that

$$\begin{align*}\deg(V) = \sum_{i=1}^m \deg(V_i), \end{align*}$$

and for each $\sigma \in \{0,1,-1\}^{\mathcal {P}}$ ,

$$\begin{align*}b_0({\mathcal R}(\sigma,V(\mathrm{R}))) \leq \sum_{i=1}^m b_0({\mathcal R}(\sigma,V_i(\mathrm{R}))), \end{align*}$$

we can reduce to the case $m=1$ and assume that $V = V_1$ .

The theorem now follows directly from Propositions 4.8, 4.9, and 4.10.

Proof. We follow closely the proof of Lemma 4.20. Using the same argument as in the proof of Lemma 4.20, we have

$$\begin{align*}\pi_{\mathrm{C}}^{-1}(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))) \cap V \subset V(\mathrm{R}). \end{align*}$$

We now prove that for each $C \in \mathrm {Cc}(V(\mathrm {R})_p)$ , there exists $D \in \mathrm {Cc}(\pi _{\mathrm {C}}(V)(\mathrm {R}) - \mathrm {Sing}(\pi _{\mathrm {C}}(V)))$ , such that $\pi ^{-1}(D) \subset C$ . It would follow that

$$\begin{align*}b_0(V(\mathrm{R})_p) \leq b_0(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))). \end{align*}$$

Let $C \in \mathrm {Cc}(V(\mathrm {R})_p)$ . Since $\pi (V(\mathrm {R})) \subset \pi _{\mathrm {C}}(V)(\mathrm {R})$ , there exists $D' \in \mathrm {Cc}(\pi _{\mathrm {C}}(V)(\mathrm {R}))$ such that $\pi (C) \subset D'$ .

Since the local real dimension of $V(\mathrm {R})_p$ is equal to p at all points of $V(\mathrm {R})_p$ , $\dim _{\mathrm {R}} C = \dim _{\mathrm {R}} D' = p$ . Moreover, since $\dim _{\mathrm {R}} (D' \cap \mathrm {Sing}(\pi _C(V))) < p$ , $D' - \mathrm {Sing}(\pi _C(V)) \neq \emptyset $ . Now let

$$\begin{align*}D \in \mathrm{Cc}(D' - \mathrm{Sing}(\pi_C(V))) \subset \mathrm{Cc}(\pi_{\mathrm{C}}(V)(\mathrm{R}) - \mathrm{Sing}(\pi_{\mathrm{C}}(V))). \end{align*}$$

Then, $\pi ^{-1}(D) \subset C$ and we are done.

Proof of Theorem 2.9.

The theorem follows from Lemma 4.22 and Lemma 4.21.

Proof of Theorem 2.12.

Let $H\subset \mathbb {P}^N_{\mathrm {C}}$ be a generic real hyperplane. Then

$$ \begin{align*} \sum_{\pi\in\{0,1\}^{\mathcal P}} b_0\big({\mathcal R}(\pi,V(\mathrm{R}))\big) \;\le\;& \sum_{\pi \in\{0,1\}^{\mathcal P}} b_0\big({\mathcal R}(\pi,\,V(\mathrm{R})\setminus H)\big) \\&\quad + \sum_{\pi\in\{0,1\}^{\mathcal P}} b_0\big({\mathcal R}(\pi,\,(V\cap H)(\mathrm{R}))\big). \end{align*} $$

Indeed, for each $\pi \in \{0,1\}^{\mathcal {P}}$ the semi-algebraic set ${\mathcal R}(\pi ,V(\mathrm {R}))$ is the disjoint union of ${\mathcal R}(\pi ,V(\mathrm {R})\setminus H)$ and ${\mathcal R}(\pi ,(V\cap H)(\mathrm {R}))$ , and every semi-algebraically connected component of ${\mathcal R}(\pi ,V(\mathrm {R}))$ contains a semi-algebraically connected component of one of these two sets.

The first sum can be bounded by the affine estimate (Theorem 2.5). For the second sum, note that the hypothesis that the local real dimension is constant on each $V_i(\mathrm {R})$ is preserved after intersecting with a generic real hyperplane; hence $V\cap H$ satisfies the same regularity assumption. Since $\dim (V\cap H)=p-1$ , we may apply the inductive hypothesis to $V\cap H$ to bound $\sum _{\pi \in \{0,1\}^{\mathcal P}} b_0\big ({\mathcal R}(\pi ,\,(V\cap H)(\mathrm {R}))\big )$ , thereby completing the induction step.

Notation 4.24. For any closed and bounded semi-algebraic set S, we denote by $b(S) = \sum _{i \geq 0} b_i(S)$ the sum of the $\mathbb {Z}/2\mathbb {Z}$ -Betti numbers of S.

Notation 4.25. Let $\mathcal {P} = \{P_1,\ldots ,P_s\} \subset \mathrm {R}[X_1,\ldots ,X_N]$ be a finite set. Let $H_1,\ldots ,H_s \in \mathrm {R}[X_1,\ldots ,X_N]$ with $H_1(x),\ldots , H_N(x)> 0$ , for all $x \in S$ , and

$$\begin{align*}\widetilde{\mathcal{P}} = \bigcup_{i=1}^{s} \{P_i \pm {\varepsilon} H_i, P_i \pm \delta H_i\} \subset \mathrm{R}{\langle}{\varepsilon},\delta{\rangle}[X_1,\ldots,X_N]. \end{align*}$$

For $\sigma \in \sigma \in \{0,1,-1\}^{\mathcal {P}}$ define $\widetilde {\sigma } \in \{0,1,-1\}^{\widetilde {\mathcal {P}}}$ as follows. For each $i, 1 \leq i \leq s$ , If $\sigma (P_i) = 0$ :

$$ \begin{align*} \widetilde{\sigma}(P_i + {\varepsilon} H_i) &= +1, \\ \widetilde{\sigma}(P_i - {\varepsilon} H_i) &= -1,\\ \widetilde{\sigma}(P_i + \delta H_i) &= +1, \\ \widetilde{\sigma}(P_i - \delta\ H_i) &= -1. \end{align*} $$

If $\sigma (P_i) = 1$ :

$$ \begin{align*} \widetilde{\sigma}(P_i + {\varepsilon} H_i) &= 1, \\ \widetilde{\sigma}(P_i - {\varepsilon} H_i) &= 1,\\ \widetilde{\sigma}(P_i + \delta H_i) &= 1, \\ \widetilde{\sigma}(P_i - \delta\ H_i) &= 1. \end{align*} $$

If $\sigma (P_i) = -1$ :

$$ \begin{align*} \widetilde{\sigma}(P_i + {\varepsilon} H_i) &= -1, \\ \widetilde{\sigma}(P_i - {\varepsilon} H_i) &= -1,\\ \widetilde{\sigma}(P_i + \delta H_i) &= -1, \\ \widetilde{\sigma}(P_i - \delta\ H_i) &= -1. \end{align*} $$

Proof. Using the fact that S is closed and bounded, it is immediate that

$$\begin{align*}\mathrm{Ext}({\mathcal R}(\sigma,S),\mathrm{R}{\langle}{\varepsilon},\delta{\rangle}) \sim {\mathcal R}(\widetilde{\sigma},\mathrm{Ext}(S,\mathrm{R}{\langle}{\varepsilon},\delta{\rangle})), \end{align*}$$

where $\sim $ denotes semi-algebraic homotopy equivalence.

Proof. It follows from the conic structure at infinity of semi-algebraic sets [Reference Bochnak, Coste and Roy11] that for all large enough $R> 0$ , for each $\sigma \in \{0,1,-1\}^{\mathcal {P}}$ , ${\mathcal R}(\sigma , V(\mathrm {R}))$ is semi-algebraically homeomorphic to ${\mathcal R}(\sigma , V(\mathrm {R})\cap B_N(0,R))$ . Let $\sigma _0$ denote the extension of the sign condition $\sigma $ to $\mathcal {P}_0 = \{P_0\} \cup \mathcal {P}$ , by setting $\sigma _0(P_0) = -1$ . Then, for all large enough $R> 0$ , ${\mathcal R}(\sigma ,V(R))$ is semi-algebraically homeomorphic to ${\mathcal R}(\sigma _0,V(\mathrm {R}))$ . Now, let $S = V(\mathrm {R}) \cap \overline {B_N(0,2R)}$ . Notice that S is a closed and bounded semi-algebraic set, and ${\mathcal R}(\sigma _0,V(\mathrm {R})) = {\mathcal R}(\sigma _0,S)$ . The lemma now follows from Lemma 4.26.

Proof. Let $K = \bigcup _{Q \in \mathcal {Q}} \mathrm {Z}(Q^h,\overline {V}(\mathrm {R}))$ . Since $\overline {V}(\mathrm {R})$ is a compact (real) manifold (orientable over $\mathbb {Z}/2\mathbb {Z}$ ), and K is a closed and bounded semi-algebraic subset, we obtain from the Lefschetz duality theorem (see, e.g., [Reference Spanier38, page 297, Theorem 18]) that for each i, $b_i(\overline {V}(\mathrm {R}) - K) = b_{\dim \overline {V} - i}(\overline {V}(\mathrm {R}),K)$ .Footnote ⁴ Moreover, using the cohomology long exact sequence, one has

$$\begin{align*}\dim \mathrm{H}^{\dim \overline{V} - i}(\overline{V}(\mathrm{R}),K) \leq \dim \mathrm{H}^{\dim \overline{V} - i}(\overline{V}(\mathrm{R})) + \dim \mathrm{H}^{\dim \overline{V} - i-1}(K). \end{align*}$$

Thus, we get

(4.7) $$ \begin{align} b_i(\overline{V}(\mathrm{R}) - K) \leq b_{\dim \overline{V} - i}(\overline{V}(\mathrm{R})) + b_{\dim \overline{V} - i-1}(K). \end{align} $$

Now suppose that $C \in \mathrm {Cc}_b({\mathcal R}(\sigma ,V(\mathrm {R}))$ for some $\sigma \in \{1,-1\}^{\mathcal {Q}}$ . Then, $C \in \mathrm {Cc}(\overline {V}(\mathrm {R}) - \bigcup _{Q \in \mathcal {Q}} \mathrm {Z}(Q^h,\overline {V}(\mathrm {R})))$ .

This implies that for each $i \geq 0$ ,

(4.8) $$ \begin{align} \sum_{\sigma \in \{1,-1\}^{\mathcal{Q}}} \sum_{C \in \mathrm{Cc}_b({\mathcal R}(\sigma,V(\mathrm{R}))} b_i(C) \leq b_i(\overline{V}(\mathrm{R}) - \bigcup_{Q \in \mathcal{Q}} \mathrm{Z}(Q^h,\overline{V}(\mathrm{R}))). \end{align} $$

The lemma now follows from inequalities (4.7) and (4.8).

Proof. The first inequality follows from the Mayer-Vietoris spectral sequence associated to the cover of the closed semi-algebraic set $S = \overline {V}(\mathrm {R}) \cap \bigcup _{i \in I} \mathrm {Z}(Q_i,\mathbb {P}^N_{\mathrm {R}})$ by the closed semi-algebraic subsets $S_i = \overline {V}(\mathrm {R}) \cap \mathrm {Z}(Q_i,\mathbb {P}^N_{\mathrm {R}})$ . The $E_1$ -page of this spectral sequence is given by

$$\begin{align*}E_1^{i,j} = \bigoplus_{H \subset [1,s], \mathrm{card}(H) = i+1} \mathrm{H}^{j}(\overline{V}(\mathrm{R}) \cap \bigcap_{h \in H} \mathrm{Z}(Q_{h},\mathbb{P}^N_{\mathrm{R}})), \end{align*}$$

with

$$\begin{align*}\mathrm{H}^n(S) = \bigoplus_{i+j=n} E_\infty^{i,j}. \end{align*}$$

The first inequality in the lemma is an immediate consequence.

Proof. If X is a compact space (or a regular complex) equipped with an involution map $c:X \rightarrow S$ , and $\mathrm {Fix}(c) \subset X$ denotes the subspace of fixed points of X, the Smith exact sequence (see, e.g., [Reference Bredon12, page 126]) then implies that

(4.9) $$ \begin{align} b(\mathrm{Fix}(c),\mathbb{Z}/2\mathbb{Z}) \leq b(X,\mathbb{Z}/2\mathbb{Z}). \end{align} $$

To prove the lemma, take the involution c to be complex conjugation.

Proof. If $\overline {V} \subset \mathbb {P}^N_{\mathrm {C}}$ is a nonsingular complete intersection variety with $\dim \overline {V} = N - \ell $ and defined as the intersection of nonsingular hypersurfaces of degrees $e_1,\ldots ,e_{\ell }$ , then $b(\overline {V})$ is given by the following classical formula (see, e.g., [Reference Eisenbud and Harris14, §5.7.2–3])

(4.10) $$ \begin{align} \nonumber b(\overline{V}) &= (1 + (-1)^{N-\ell+1})\cdot (N-\ell+1) \\\nonumber & \quad + e_1 \cdot e_2\cdots e_\ell\cdot \left(\sum_{i=0}^{N-\ell}(-1)^{i}\cdot\binom{N+1}{i} \cdot h_{N-\ell -i}(e_1,\ldots,e_\ell) \right) \\&= (1 + (-1)^{N-\ell+1})\cdot (N-\ell+1) \\\nonumber & \quad + e_1 \cdot e_2\cdots e_\ell\cdot \left(\sum_{j=0}^{N-\ell}(-1)^{N-\ell -j}\cdot\binom{N+1}{j+\ell+1} \cdot h_{j}(e_1,\ldots,e_\ell) \right), \end{align} $$

where $h_j(e_1,\ldots ,e_\ell )$ is the complete homogeneous symmetric polynomial of degree j in $(e_1,\ldots , e_\ell )$ .

In our case, we can assume that the degree sequence of the polynomials defining $\overline {V}$ is of length $\ell = N-p+q$ . Suppose that the degree sequence is $e_1,\ldots ,e_{N-p+q}$ . We can assume without loss of generality that $e_1 \cdots e_{N-p} =D$ , and $e_{N-p+1} \leq d,\ldots ,e_{N-p+q} \leq d$ .

Thus,

(4.11) $$ \begin{align} e_1 \cdots e_\ell \leq D \cdot d^q, \end{align} $$

and

(4.12) $$ \begin{align} \nonumber \left(\sum_{j=0}^{N-\ell}(-1)^{N-\ell -j}\cdot\binom{N+1}{j+\ell+1} \cdot h_{j}(e_1,\ldots,e_\ell) \right) &\leq 2^{N+1} h_{N-\ell}(e_1,\ldots,e_\ell) \\ &\leq 2^{N+1} d^{N-\ell} \\ \nonumber &= 2^{N+1} d^{p-q}. \end{align} $$

The lemma now follows from the equality (4.10), and the inequalities (4.11) and (4.12).

Proof of Theorem 2.14.

Let $\overline {V} \subset \mathbb {P}_{\mathrm {C}}^N$ be a nonsingular complete intersection real variety with ${\deg (V) = D}$ and $\dim (V) = p$ , and $V = \overline {V} \cap \mathrm {C}^N$ .

Let $\mathcal {P} \subset \mathrm {R}[X_1,\ldots ,X_N]_{\leq d}$ be a finite set of polynomials with $d \geq D$ and $\mathrm {card}(\mathcal {P}) = s$ .

First, observe that it follows from the theorem of conic structure at infinity [Reference Basu, Pollack and Roy5, Proposition 5.49] that for all large enough $R> 0$ ,

$$\begin{align*}\sum_{\sigma \in \{0,1,-1\}^{\mathcal{P}}} b_0({\mathcal R}(\sigma,V(\mathrm{R}))) \leq \sum_{\sigma \in \{0,1,-1\}^{\mathcal{P}}} b_0({\mathcal R}(\sigma,V(\mathrm{R}) \cap \overline{B_N(0,R)})). \end{align*}$$

Now, let $\mathcal {P}_0 = \mathcal {P} \cup \{\sum _{i=1}^N X_i^2 - R\}$ . We have that

$$ \begin{align*} \sum_{\sigma \in \{0,1,-1\}^{\mathcal{P}}} b_i({\mathcal R}(\sigma,V(\mathrm{R}))) &\leq \sum_{\sigma \in \{0,1,-1\}^{\mathcal{P}}} b_i({\mathcal R}(\sigma,V(\mathrm{R}) \cap \overline{B_N(0,R)})) \\ &\leq \sum_{\sigma' \in \{0,1,-1\}^{\mathcal{P}_0}, C \in \mathrm{Cc}_b({\mathcal R}(\sigma',V(\mathrm{R})))} b_i(C) \\ &\leq \sum_{\widetilde{\sigma} \in \{1,-1\}^{\widetilde{\mathcal{P}}}} b_i({\mathcal R}(\widetilde{\sigma},V(\mathrm{R}{\langle}{\varepsilon},\delta{\rangle}))) \text{ (using Lemma~}4.27) \\&\leq b(\overline{V}(\mathrm{R})) + b_{p-i-1}(\bigcup_{\widetilde{P} \in \widetilde{\mathcal{P}}} \mathrm{ Z}(\widetilde{P}^h,\overline{V}(\mathrm{R}{\langle}{\varepsilon},\delta{\rangle}))) \\ &\text{ (using Lemma~}4.28). \end{align*} $$

The theorem now follows from Lemmas 4.29, 4.30, and 4.31.

Proof. Similar to the proof of Lemma 4.26 and omitted.

Proof. For $1 \leq i \leq N-p$ , let $H_1,\ldots ,H_{N-p}$ be generically chosen polynomials with $\deg (H_i) = \deg (Q_i)$ . Now, let $\widetilde {Q}_i = (1-{\varepsilon })Q_i + {\varepsilon } H_i$ . Then, properties (A) and (B) follow from the fact that the same holds for the polynomials $H_1,\ldots ,H_{N-p}$ and a standard transfer argument.

For property (C), first observe that clearly

$$\begin{align*}\lim_{\varepsilon} \mathrm{Z}(\widetilde{\mathcal{Q}},B_{2N}(0,R+1)) \subset \mathrm{Z}(\mathcal{Q},\overline{B_{2N}(0,R+1)}) \end{align*}$$

from which it follows that

$$\begin{align*}\lim_{\varepsilon} \mathrm{Z}(\widetilde{\mathcal{Q}},B_{2N}(0,R+1)) \cap \overline{B_{2N}(0,R)} = \mathrm{ Z}(\mathcal{Q},\overline{B_{2N}(0,R)}). \end{align*}$$

To prove the reverse inclusion, let $x \in (\mathcal {Q},\overline {B_{2N}(0,R)})$ . We prove that there exists $\widetilde {x} \in \mathrm { Z}(\widetilde {\mathcal {Q}},B_{2N} (0,R+1))$ such that $\lim _{\varepsilon } \widetilde {x} = x$ .

Let $\mathbb {P}^{N-p}_{\mathrm {C}} \subset \mathbb {P}^{N}_{\mathrm {C}}$ be a generic $(N-p)$ -dimensional plane through x. Then, using Bezout’s theorem, $\mathrm {Z}(\mathcal {Q}^h,\mathbb {P}^{N-p}_{\mathrm {C}})$ contains x and has $d_1 \cdots d_{N-p}$ points counted with multiplicities. But $\mathrm {Z}(\widetilde {\mathcal {Q}}^h,\mathbb {P}^{N-p}_{\mathrm {C}})$ also contains $d_1 \cdots d_{N-p}$ points each of multiplicity one, and moreover $\lim _{\varepsilon } \mathrm {Z}(\widetilde {\mathcal {Q}}^h,\mathbb {P}^{N-p}_{\mathrm {C}}) = \mathrm { Z}(\widetilde {\mathcal {Q}}^h,\mathbb {P}^{N-p}_{\mathrm {C}})$ . From this it follows that there must exist $\widetilde {x} \in \mathrm {Z}(\widetilde {\mathcal {Q}}^h,\mathbb {P}^{N-p}_{\mathrm {C}})$ such that $\lim _{\varepsilon } \widetilde {x} = x$ , and clearly ${\widetilde {x} \in \mathrm {Z}(\widetilde {\mathcal {Q}},B_{2N}(0,R+1))}$ .

Proof. This follows immediately from Lemma 4.32, noting that ${\mathcal R}(\sigma ,\mathrm {C}^N)$ is an open semi-algebraic subset.

Proof. This follows immediately from Lemma 4.34.

Proof of Theorem 2.17.

Using Lemma 4.35, it suffices to bound

$$\begin{align*}\sum_{\widetilde{\sigma} \in \{1,-1\}^{\widetilde{\mathcal{P}}}} b_0({\mathcal R}(\widetilde{\sigma},\widetilde{V})). \end{align*}$$

Now for $j=1,\ldots ,N-p$ , let

$$\begin{align*}\widetilde{Q}_j = \widetilde{Q}_{j,0} + \sqrt{-1}\cdot \widetilde{Q}_{j,1}, \end{align*}$$

where $\widetilde {Q}_{j,0}, \widetilde {Q}_{j,1} \in \mathrm {R}[X_1,Y_1,\ldots ,X_n,Y_n]$ have generic coefficients.

Thus,

$$\begin{align*}\widetilde{Q}_{1,0},\widetilde{Q}_{1,1},\ldots, \widetilde{Q}_{N-p,0},\widetilde{Q}_{N-p,1} \end{align*}$$

define a nonsingular complete intersection in $\widetilde {W} \subset \mathrm {C}^{2N}$ , with $\deg (\widetilde {W}) = D^2$ and $\dim \widetilde {W} = 2p$ , and with $\widetilde {W}(\mathrm {R}) = \widetilde {V}$ .

Now apply Theorem 2.14 with a nonsingular complete intersection $\widetilde {W}$ and the family of polynomials $\widetilde {\mathcal {P}}$ to obtain

$$ \begin{align*} \sum_{\widetilde{\sigma} \in \{1,-1\}^{\widetilde{\mathcal{P}}}} b_0({\mathcal R}(\widetilde{\sigma},\widetilde{V})) &\leq \sum_{\sigma \in \{0,1,-1\}^{\widetilde{\mathcal{P}}}} b_0({\mathcal R}(\widetilde{\sigma},\widetilde{W}(\mathrm{R}))) \\ &\leq (D^2)^2 \cdot (O(s d))^{2 p} \\ &= D^4 \cdot (O(s d))^{2 p}.\\[-37pt] \end{align*} $$

Proof of Theorem 2.19.

Using the conic structure at infinity for semi-algebraic sets (see, e.g., [Reference Basu, Pollack and Roy5, Proposition 5.49]), choose $R> 0, R \in \mathrm {R}$ , such that for each $\sigma \in \{0,1,-1\}^{\mathcal {P}}$ , ${\mathcal R}(\sigma ,V(\mathrm {R}))$ is semi-algebraically homeomorphic to ${\mathcal R}(\sigma ,V(\mathrm {R}) \cap B_N(0,R))$ . We denote $\mathcal {P}_0 = \mathcal {P} \cup P_0$ , where $P_0 = \sum _{i=1}^N X_i^2 - R$ .

Applying Lemma 4.27 (using the same notation as in Lemma 4.27), we have

$$ \begin{align*} \sum_{\sigma \in \{0,1-1\}^{\mathcal{P}}} b_i({\mathcal R}(\sigma,V(\mathrm{R}))) \leq \sum_{\widetilde{\sigma} \in \{1,-1\}^{\widetilde{P}_0}} b_i({\mathcal R}(\widetilde{\sigma}),\mathrm{Ext}(V(\mathrm{R}),\widetilde{\mathrm{R}}), \end{align*} $$

and finally, using Lemma 4.28, that $\sum _{\widetilde {\sigma } \in \{1,-1\}^{\widetilde {P}_0}} b_i({\mathcal R}(\widetilde {\sigma }),\mathrm { Ext}(V(\mathrm {R}),\widetilde {\mathrm {R}}))$ is bounded by

$$\begin{align*}b_{p-i-1}\left(\bigcup_{\widetilde{P} \in \widetilde{\mathcal{P}}_0} \mathrm{Z}(\widetilde{P}^h, \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}})) \cup \mathrm{Z}(X_0, \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}})) \right) + b_{p-i}( \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}})). \end{align*}$$

We now use the Mayer-Vietoris spectral sequence to observe that the right-hand side of the previous equation is bounded by the sum of three terms (we use the convenient notation $A \subset _n B$ to denote $A \subset B$ and $\mathrm {card}(A) = n$ ):

1. A.

   $$\begin{align*}\sum_{0 \leq j \leq p-i-1} \sum_{\widetilde{\mathcal{P}}_0' \subset_{j+1} \widetilde{\mathcal{P}}_0} b_{p-i-1-j}(\mathrm{ Z}((\widetilde{\mathcal{P}}_0')^h, \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}}))), \end{align*}$$

2. B.

   $$\begin{align*}\sum_{0 \leq j \leq p-i-2} \sum_{\widetilde{\mathcal{P}}_0' \subset_{j+1} \widetilde{\mathcal{P}}_0} b_{p-i-1-j}(\mathrm{Z}(\{X_0\} \cup (\widetilde{\mathcal{P}}_0')^h, \mathrm{ Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}}))), \end{align*}$$

3. C.

   $$\begin{align*}b( \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}})). \end{align*}$$

We are now going to bound from above each one of the three items. Firstly, observe that, using Remark 1.5 and the fact that V is nonsingular, we can bound the quantity in Part (C) by $D^{p+1}$ .

We now bound for each $\widetilde {\mathcal {P}}_0' \subset _{j+1} \widetilde {\mathcal {P}}_0$ , the Betti number

$$\begin{align*}b_{p-i-1-j}(\mathrm{Z}((\widetilde{\mathcal{P}}_0')^h, \mathrm{Ext}(\overline{V}(R),\widetilde{\mathrm{R}}))). \end{align*}$$

Note that $\mathrm {Z}((\widetilde {\mathcal {P}}_0')^h, \mathrm {Ext}(\overline {V}(\mathrm {R}),\widetilde {\mathrm {R}}))$ is the real part of a nonsingular projective hypersurface with degree bounded by $D d^{j+1}$ and dimension $p-j -1$ . Using Remark 1.5 and the Thom-Smith inequality will give us

$$\begin{align*}b_{p-i-1-j}(\mathrm{Z}((\widetilde{\mathcal{P}}_0')^h, \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}}))) \leq (D d^{j+1})^{p-j} = D^{p-j} d^{(j+1)(p-j)}. \end{align*}$$

Using perturbation techniques, we prove a slightly better bound as follows.

Let $S = \mathrm {Z}((\widetilde {\mathcal {P}}_0')^h, \mathrm {Ext}(\overline {V}(\mathrm {R}),\widetilde {\mathrm {R}}))$ .

First let

$$\begin{align*}\widetilde{Q} = \sum_{\widetilde{P} \in \widetilde{\mathcal{P}}_0'} \widetilde{P}^2. \end{align*}$$

Let $H \in \widetilde {\mathrm {R}}[X_0,\ldots ,X_N]$ be a generic homogeneous polynomial of the same degree as $\widetilde {Q}$ and which is nonnegative in $\mathrm {R}^{N+1}$ . Note that in $\mathbb {P}_{\mathrm {R}}^N$ the sign of a homogeneous polynomial of even degree is well defined. It follows from [Reference Basu, Pollack and Roy5, Lemma 16.17] that the semi-algebraic set $\widetilde {S} \subset \mathrm {Ext}(\overline {V}(R),\widetilde {R}{\langle }\zeta {\rangle }))$ defined as the intersection of $\mathrm {Ext}(\overline {V}(R),\widetilde {R}{\langle }\zeta {\rangle }))$ with the semi-algebraic set defined by $(1-\zeta )\widetilde {Q} - \zeta H \leq 0$ is semi-algebraically homotopy equivalent to $ \mathrm {Ext}(S,\widetilde {\mathrm {R}}{\langle }\zeta {\rangle })$ .

Moreover, it follows from the Mayer-Vietoris exact sequence that

$$\begin{align*}b(\widetilde{S}) \leq b(\mathrm{Z}((1-\zeta)\widetilde{Q} - \zeta H, \mathrm{ Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}}{\langle}\zeta{\rangle}))) + b(V(\mathrm{R})). \end{align*}$$

Now observe that $\mathrm {Z}((1-\zeta )\widetilde {Q} - \zeta H, \mathrm { Ext}(\overline {V}(\mathrm {R}),\widetilde {\mathrm {R}}{\langle }\zeta {\rangle }))$ is the real part of a nonsingular projective variety of degree bounded by $2 D d$ and dimension $p-1$ . It follows (using Remark 1.5 and the Thom-Smith inequality again) that

$$\begin{align*}b(\mathrm{Z}((1-\zeta)\widetilde{Q} - \zeta H, \mathrm{Ext}(\overline{V}(\mathrm{R}),\widetilde{\mathrm{R}}{\langle}\zeta{\rangle}))) \leq (2 D d)^{p}. \end{align*}$$

Thus, the sum in Part (A) is bounded by

$$\begin{align*}\left(\sum_{j=0}^{p-i-1} \binom{s}{j+1}\right)(D^p(2d)^{p} + D). \end{align*}$$

We use the same method to bound the sum in Part (B), noting that all subvarieties are contained in the $\mathbb {P}_{\mathrm {R}}^{N-1}$ defined by $X_0=0$ . This yields that the sum in Part (B) is bounded by

$$\begin{align*}\left(\sum_{j=0}^{p-i-2} \binom{s}{j+1}\right)(D^{p-1}(2d)^{p-1} + D). \end{align*}$$

Finally, summing the three parts together, we obtain that

$$ \begin{align*} \sum_{\sigma \in \{0,1-1\}^{\mathcal{P}}} b_i({\mathcal R}(\sigma,V(\mathrm{R}))) &\leq \left(\sum_{j=0}^{p-i-1} \binom{s}{j+1}\right)(D^p(2d)^{p} + D) \\& \quad + \left(\sum_{j=0}^{p-i-2} \binom{s}{j+1}\right)(D^{p-1}(2d)^{p-1} + D) \\& \quad + D^{p+1} \\&\leq O(1)^p s^{p-i} D^p(d^p + D).\\[-37pt] \end{align*} $$