Computing the Homology of Semialgebraic Sets. I: Lax Formulas

Abstract

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. All previous algorithms solving this problem have doubly exponential complexity. Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.

A On the Smoothness Assumption in Thom’s First Isotopy Lemma

We begin observing that we can define Whitney stratifications of any subset of a manifold in the same manner we define Whitney stratifications of the manifold itself.

The following lemma will be instrumental in our proof.

Lemma A.1

Let $\mathcal {M}$ be a smooth manifold and $\mathcal {S}$ be a locally finite partition of a locally closed subset $\Omega \subset \mathcal {M}$. Then:

• 1. Let $\mathcal {S}^c$ be the partition whose elements are the connected components of the elements in $\mathcal {S}$. If $\mathcal {S}$ is a Whitney stratification, then so is $\mathcal {S}^c$.

• 2. If $\mathcal {S}$ is a Whitney stratification with connected strata, then it satisfies the boundary condition:

  $$\begin{aligned} \text{ for } \sigma ,\varsigma \in \mathcal {S}, \text{ if } \varsigma \cap \overline{\sigma }\ne \varnothing , \text{ then } \varsigma \subseteq \overline{\sigma }. \end{aligned}$$

  (BC)

• 3. Let $\mathcal {S}$ satisfy the boundary condition (BC). Then, $\mathcal {S}$ is a Whitney stratification if and only if for all $\sigma \in \mathcal {S}$,

  $$\begin{aligned} \mathcal {S}_{|\overline{\sigma }}:=\{\varsigma \in \mathcal {S}\mid \varsigma \subseteq \overline{\sigma } \} \end{aligned}$$

  is a Whitney stratification of $\overline{\sigma }$.

• 4. Let $\mathcal {S}$ be a Whitney stratification, $\{\sigma _i\}_{i\in I}\subseteq \mathcal {S}$ a family of strata of the same dimension and $\mathcal {S}'$ the partition obtained from $\mathcal {S}$ by replacing the $\sigma _i$ by its union. Then, $\mathcal {S}'$ is a Whitney stratification.

Proof

Parts 1 and 2 are [35, Ch. II, Theorem 5.6 and Corollary 5.7], respectively. For part 3, the fact that $\mathcal {S}$ satisfies (BC) implies that $\mathcal {S}_{|\overline{\sigma }}$ is a partition of $\overline{\sigma }$ whose elements are elements in $\mathcal {S}$. As a subset of a Whitney stratification is a Whitney stratification itself we have shown the “only if” part. We next show the converse. For every $\sigma \in \mathcal {S}$, the fact that $\mathcal {S}_{|\overline{\sigma }}$ is a Whitney stratification implies that $\sigma \in \mathcal {S}_{|\overline{\sigma }}$ is a locally closed smooth submanifold. Next note that Whitney’s condition b needs to be checked only for pairs $(\sigma ,\varsigma )\in \mathcal {S}^2$ such that $\varsigma \cap \overline{\sigma }\ne \varnothing$. But the fact that $\mathcal {S}$ satisfies (BC) implies that, for any such pair, $\sigma ,\varsigma \in \mathcal {S}_{|\overline{\sigma }}$ and therefore, it satisfies condition b because, by hypothesis, $\mathcal {S}_{|\overline{\sigma }}$ is a Whitney stratification.

We finally prove 4. By the local character of Definition 4.1 of Whitney stratification, it is enough to check the conditions in this definition in some open neighborhood $U_x$ around each point $x\in \Omega$. Since $\mathcal {S}$ is locally finite, we can pick each $U_x$ such that $\mathcal {S}_{|U_x}$ is finite. Hence, without loss of generality, we can assume that I is finite.

For all $i\ne j\in I$ we have $\sigma _i\cap \overline{\sigma _j}=\varnothing$. Otherwise, by [35, Ch. I, (1.1)], we would have $\dim \sigma _i< \dim \sigma _j$, contradicting our hypothesis. Hence, for all $i\in I$, there is an open set $U_i$ such that $\sigma _i\subseteq U_i$ and, for all $j\ne i$, $U_i\cap \overline{\sigma _j}=\varnothing$. It follows that $\cup \sigma _i$ is a locally closed smooth manifold. The verification of the conditions in Definition 4.1 for $\mathcal {S}'$ is now straightforward. $\square$

To prove Theorem 4.4, we will rely on the following version of Thom’s First Isotopy Lemma which is the one in [35, Ch. II, Theorem 5.2].

Theorem A.2

Let $\mathcal {M}$ be a smooth manifold and $\Omega \subseteq \mathcal {M}$ a locally closed subset with a Whitney stratification $\mathcal {S}$ and let $\alpha :\mathcal {M}\rightarrow \mathbb {R}^k$ be a smooth proper map such that:

• for each stratum $\sigma \in \mathcal {S}$, $\alpha _{|\sigma }:\sigma \rightarrow \mathbb {R}^k$ is surjective,

• for each stratum $\sigma \in \mathcal {S}$, $\alpha _{|\sigma }:\sigma \rightarrow \mathbb {R}^k$ is a smooth submersion.

Then, $\alpha _{|X}$ is a trivial fiber bundle. $\square$

To deduce Theorem 4.4 from this result, we will employ graphs of maps. This will allow us to transform our not necessarily smooth map into a smooth one, as it will be simply a projection.

Let A and B be smooth manifolds. Recall that the graph of a function $\varphi :A\rightarrow B$ is the set

$$\begin{aligned} \Gamma _\varphi :=\{(a,b)\in A\times B\mid \varphi (a)=b\}. \end{aligned}$$

Associated with the graph, we have the functions $i_\varphi :A\rightarrow \Gamma _\varphi$, given by $a\mapsto (a,\varphi (a))$, and $\pi :A\times B\rightarrow B$, given by $(a,b)\mapsto b$. Clearly, $\varphi =\pi \circ i_\varphi$. Also, it is easy to see, if $\varphi$ is a continuous map, then $\Gamma _\varphi$ is a closed subset of $A\times B$ and $i_\varphi$ is a homeomorphism between A and $\Gamma _\varphi$. Finally, if $\varphi$ is a smooth map, then $\Gamma _\varphi$ is a closed smooth submanifold of $A\times B$ and $i_\varphi$ is a diffeomorphism between A and $\Gamma _\varphi$. Given a subset $X\subseteq A$, we will consider

$$\begin{aligned} \Gamma _\varphi (X):=\Gamma _{\varphi _{|X}}= \{(a,b)\in \Gamma _\varphi \mid a\in X\}= \{(a,\varphi (a))\mid a\in X\}. \end{aligned}$$

It is again clear that if $\varphi$ is continuous and X is a locally closed subset of A, then $\Gamma _\varphi (X)$ is a locally closed subset of $A\times B$. Moreover, if $\varphi$ is smooth and X is a locally closed smooth submanifold of A, then $\Gamma _\varphi (X)$ is a locally closed smooth submanifold of $A\times B$.

Proof of Theorem 4.4

Consider the graph $\Gamma _{\alpha }$ of $\alpha$. Although not necessarily a manifold (as $\alpha$ may be non-smooth), it is a locally closed subset of $\mathcal {M}\times \mathbb {R}^k$. Next consider the partition of $\Gamma _\alpha$ given by

$$\begin{aligned} \Gamma _\alpha (\mathcal {S}):= \{\Gamma _\alpha (\sigma )\mid \sigma \in \mathcal {S}\} \end{aligned}$$

and its associated partition $\Gamma ^c_\alpha (\mathcal {S})$ as defined in Lemma A.1(1).

We claim that $\Gamma ^c_\alpha (\mathcal {S})$ is a Whitney stratification of $\Gamma _\alpha$.

To prove the claim we first observe that $\Gamma ^c_\alpha (\mathcal {S})=\Gamma _\alpha (\mathcal {S}^c)$. As, by Lemma A.1(1), $\mathcal {S}^c$ is a Whitney stratification and, by construction, has connected strata, Lemma A.1(2) shows that it satisfies the boundary condition (BC). It follows that $\Gamma ^c_\alpha (\mathcal {S})$ satisfies (BC) as well.

Let $\sigma \in \mathcal {S}^c$ and $\sigma '\in \mathcal {S}$ such that $\sigma \subseteq \sigma '$. By the first hypothesis in our statement, there is an open neighborhood U of $\overline{\sigma '}\supseteq \overline{\sigma }$ and a smooth map $\varphi :U\rightarrow \mathbb {R}^k$ such that $\alpha _{|\sigma '}=\varphi$. Clearly, $\alpha _{|\sigma }=\varphi$ as well. This implies that $\Gamma _\alpha (\mathcal {S}^c_{|\overline{\sigma }}) =\Gamma _\varphi (\mathcal {S}^c_{|\overline{\sigma }})$ and $\Gamma _\alpha (\overline{\sigma }) =\Gamma _\varphi (\overline{\sigma })$. Since $\varphi$ is smooth, $\Gamma _\varphi$ is a locally closed smooth submanifold and $i_\varphi :U\rightarrow \Gamma _\varphi$ is a diffeomorphism mapping the Whitney stratification $\mathcal {S}^c_{|\overline{\sigma }}$ to $\Gamma ^c_\alpha (\mathcal {S}_{|\overline{\sigma }})$ and the closed set $\overline{\sigma }$ to $\Gamma _\alpha (\overline{\sigma })$. Hence, by [35, Ch. I, (1.4)], $\Gamma ^c_\alpha (\mathcal {S}_{|\overline{\sigma }})$ is a Whitney stratification of $\Gamma _\alpha (\overline{\sigma })$.

As this happens for all strata $\Gamma ^c_\alpha (\overline{\sigma })$ of the partition $\Gamma ^c_\alpha (\mathcal {S})$ we may apply Lemma A.1(3) to deduce that $\Gamma ^c_\alpha (\mathcal {S})$ is a Whitney stratification. We finally apply Lemma A.1(4) (several times for each dimension) to deduce that $\Gamma _\alpha (\mathcal {S})$ itself is a Whitney stratification. This proves the claim.

Since $\Gamma _\alpha (\mathcal {S})$ is a Whitney stratification of $\Gamma _\alpha$ the map $i_{\alpha }$ restricts to a diffeomorphism between $\sigma$ and $\Gamma _\alpha (\sigma )$, for all $\sigma \in \mathcal {S}$. In addition, as $\alpha =\pi \circ i_\alpha$, we have $\alpha _{|\sigma }=\pi _{|\Gamma _\alpha (\sigma )}\circ (i_\alpha )_{|\sigma }$ and, hence, as $(i_\alpha )_{|\sigma }$ is a diffeomorphism, $\pi _{|\Gamma _\alpha (\sigma )}$ is surjective if and only if $\alpha _{|\sigma }$ is so, and $\pi _{|\Gamma _\alpha (\sigma )}$ is a smooth submersion if and only if so is $\alpha _{|\sigma }$. In summary, the last two hypotheses of our statement imply the hypothesis of Theorem A.2, and consequently, that $\pi _{|\Gamma _\alpha }$ is a trivial bundle.

We can now conclude because a trivialization $h:\Gamma _\alpha \rightarrow F\times \mathbb {R}^k$ of $\pi _{|\Gamma _\alpha }$ induces the trivialization $h\circ i_\alpha$ of $\alpha :\mathcal {M}\rightarrow \mathbb {R}^k$. $\square$