On a Nash curve selection lemma through finitely many points

Abstract

A celebrated theorem in real algebraic and analytic geometry (originally due to Bruhat–Cartan and Wallace, and stated later in its current form by Milnor) is the (Nash) curve selection lemma, which has wide applications. It states that each point in the closure of a semialgebraic set $\mathcal{S}\subset {\mathbb{R}}^n$ can be reached by a Nash arc of ${\mathbb{R}}^n$ such that at least one of its branches is contained in $\mathcal{S}$.
The purpose of this work is to generalize the previous result to finitely many points. More precisely, let $\mathcal{S}\subset {\mathbb{R}}^n$ be a semialgebraic set, let $x_1,\dots ,x_r\in \mathcal{S}$ be $r$ points (that we call ‘control points’) and let $0=:t_1<\cdots <t_r:=1$ be $r$ values (that we call ‘control times’). A natural ‘logistic’ question concerns the existence of a smooth and semialgebraic (Nash) path $\alpha :[0,1]\to \mathcal{S}$ that passes through the control points at the control times, that is, $\alpha (t_k)=x_k$ for $k=1,\dots ,r$. The necessary and sufficient condition to guarantee the existence of $\alpha$ when the number of control points is large enough and they are in general position is that $\mathcal{S}$ is connected by analytic paths. The existence of generic real algebraic sets that do not contain rational curves confirms that the analogous result involving polynomial paths (instead of Nash paths) is only possible under additional restrictions. A sufficient condition is that $\mathcal{S}\subset {\mathbb{R}}^n$ has, in addition, dimension $n$.
A related problem concerns the approximation by a Nash path of an existing continuous semialgebraic path $\beta :[0,1]\to \mathcal{S}$ with control points $x_1,\dots ,x_r\in \mathcal{S}$ and control times $0=:t_1<\cdots <t_r:=1$. As one can expect, apart from the restrictions on $\mathcal{S}$, some restrictions on $\beta$ are needed. A sufficient condition is that the (finite) set of values $\eta (\beta )$ at which $\beta$ is not smooth is contained in the set of regular points of $\mathcal{S}$ and $\eta (\beta )$ does not meet the set of control times.
If $\mathcal{S}\subset {\mathbb{R}}^n$ is a finite union (connected by analytic paths) of $n$-dimensional convex polyhedra, we can even ‘estimate’ (using Bernstein’s polynomials) the degree of the involved polynomial path. This requires: (1) a polynomial double curve selection lemma for convex polyhedra involving only degree $3$ cuspidal curves; (2) to find the simplest polynomial paths that connect two convex polyhedra (whose union is connected by analytic paths), and (3) some improvements concerning well-known bounds for Bernstein’s polynomials (and their high order derivatives) to approximate continuous functions that are not differentiable on their whole domain.