# Rolle sets

Let $\R$ be an o-minimal expansion of the real field.

Definition
A Rolle leaf over $\R$ is a Rolle leaf of a definable $(n-1)$-distribution on $\RR^n$, for some $n \in \NN$.

Example
The graph of $\exp$ is a Rolle leaf over $\bar\RR$.

Exercise
Let $C \subseteq M^n$ be a definable $C^1$-cell of dimension $m$, let $d$ be a definable $(m-1)$-distribution on $C$ and $L$ be a Rolle leaf of $d$. Show that $L$ is the projection of a Rolle leaf over $\R$.

We let $\R_1$ be the expansion of $\R$ by all Rolle leaves over $\R$; our next goal is to prove that $\R_1$ is o-minimal. The quantifier-free definable sets of $\R_1$ are the following:

Definition
A basic Rolle set over $\R$ is a set of the form $A \cap L_1 \cap \cdots \cap L_k$, where $A$ is definable and each $L_i$ is a Rolle leaf over $\R$. A Rolle set over $\R$ is a finite union of basic Rolle sets over $\R$.

The goal of this post is to collect all the properties of Rolle sets needed to establish o-minimality of $\R_1$.

Exercise 11: Properties 1 through 4
Show that the collection of all Rolle sets over $\R$ is closed under taking finite intersections, finite unions, cartesian products and permutations of coordinates.

Property 5
Let $W \subseteq \RR^n$ be a Rolle set over $\R$. Then there exists a closed Rolle set $V \subseteq \RR^{n+1}$ over $\RR$ such that $W = \Pi_n(V)$.

Proof. Let $A \subseteq \RR^n$ be definable and $L_1, \dots, L_k \subseteq \RR^n$ be Rolle leaves over $\R$ such that $W = A \cap L_1 \cap \cdots \cap L_k$. By cell decomposition, we may assume that $A$ is a cell; in particular, the frontier $\fr A$ is a definable closed set. Therefore, the function $d:A \into (0,\infty)$ defined by $$d(x):= 1/\dist(\fr A,x)$$ is definable and continuous, and its graph is a closed, definable subset of $\RR^{n+1}$ such that $\Pi_n(\gr(d)) = A$.

On the other hand, each $L_i’:= L_i \times \RR$ is a Rolle leaf over $\R$; since Rolle leaves are closed, it follows that $$V:= \gr(d) \cap L_1′ \cap \cdots \cap L_k’$$ is a closed Rolle set over $\R$. By the definition of $L_i’$, we have $W = \Pi_n(V)$, as required. $\qed$

Property 6
Let $W \subseteq \RR^n$ be a Rolle set over $\R$ and $m \le n$. Then there exists $K \in \NN$ such that, for any $a \in \RR^m$, the fiber $W_a$ of $W$ has at most $K$ connected components.

Proof. Note that, for $a \in \RR^m$ and $i = 1, \dots, m$, the hyperplane $$V_i:= \set{x \in \RR^n:\ x_i = a_i}$$ is a Rolle leaf of the definable distribution $e_i:= \ker(dx_i)$, and that $$\Pi_m^{-1}(\{a\}) = V_1 \cap \cdots \cap V_m.$$ Since $W_a$ is homeomorphic to $W \cap \Pi_m^{-1}(\{a\})$, Property 6 now follows from Khovanskii theory; the details are left as an exercise. $\qed$

For the last property, we need a definition:

Definition
A manifold $M \subseteq \RR^n$ is in standard position if, for every strictly increasing $\iota:\{1, \dots, m\} \into \{1, \dots, n\}$, the restriction of $\Pi_\iota$ to $M$ has constant rank.

Remark
Let $M \subseteq \RR^n$ be a manifold in standard position, $\iota:\{1, \dots, m\} \into \{1, \dots, n\}$ be strictly increasing and $a \in \RR^m$. Then, by the constant rank theorem for manifolds (see for instance Theorem 10 and Proposition 12 in Chapter 2 of Spivak’s book), the set $$M_{\iota,a}:= M \cap \Pi_{\iota}^{-1}(\{a\})$$ is either empty of a manifold of dimension $\dim M – r$, where $r$ is the rank of $\Pi_\iota\rest{M}$, and the projection $\Pi_\iota(M)$ is a countable union of submanifolds of $\RR^m$ of dimension $r$.

Exercise
Show that the manifold $M_{\iota,a}$ of the previous remark is a manifold in standard position.

Proposition
Let $A \subseteq \RR^n$ be definable and $d_1, \dots, d_k$ be definable $(n-1)$-distributions on $\RR^n$. Then, for $p \in \NN$, there is a finite partition $\P$ of $A$ into definable $C^p$-cells such that, whenever $N \in \P$ and $L_i$ is a Rolle leaf of $d_i$ for each $i$, the set $N \cap L_1 \cap \cdots \cap L_k$ is a manifold in standard position.

Proof. By Proposition 1 of this post, there is a finite partition $\P$ of $A$ into definable $C^1$-cells compatible with $d_1, \dots, d_k$, as well as the $(n-1)$-distributions $e_1, \dots, e_n$ defined by $$e_i:= \ker(dx_i).$$ Now let $N \in \P$ and fix Rolle leaves $L_i$ of $d_i$, for $i=1, \dots, k$; we need to show that $$M:= N \cap L_1 \cap \cdots \cap L_k$$ is a manifold in standard position. So let $\iota:\{1, \dots, m\} \into \{1, \dots, n\}$ be strictly increasing and $a \in \RR^m$, and write $$d:= d_1 \cap \cdots \cap d_k \cap e_1 \cap \cdots \cap e_n.$$ Our choice of $\P$ implies that the distribution $d^N$ induced on $N$ by $d$ has dimension $p$, say; in particular, the set $M_{\iota,a}$ is a submanifold of $N$ of dimension $p$. Note that $p$ is independent of $a$.

Moreover, for $x \in M_{\iota,a}$, we have $$\ker\left(\Pi_{\iota}\rest{T_xM}\right) = T_xM \cap \ker(\Pi_{\iota})$$ $$= g_N(x) \cap d_1(x) \cap \cdots \cap d_k(x) \cap e_1(x) \cap \cdots \cap e_n(x)$$ $$=d^N(x).$$ Therefore, $\ker\left(\Pi_{\iota}\rest{T_xM}\right)$ has constant dimension $p$, for all $a \in \RR^m$ and $x \in M_{\iota,a}$, which implies (by linear algebra) that $\Pi_{\iota} \rest{M}$ has constant rank. $\qed$

As a corollary of this proposition, we obtain:

Property 7
Every Rolle set over $\R$ is a finite union of Rolle sets over $\R$ that are manifolds in standard position. $\qed$

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