On a Nash curve selection lemma through finitely many points
José F. Fernando
TL;DR
The article advances the Nash curve selection theory by providing a constructive solution to logistic problems with finitely many control points in semialgebraic sets, under the assumption that the set is connected by analytic paths. It develops a robust framework combining Nash curves, polynomial-bridge constructions, and Bernstein-approximation techniques to produce Nash (and in PL cases polynomial) paths through prescribed points while staying inside the prescribed regions. The work delivers a general main theorem (Smart Nash curve) with a PL specialization and a graph-based interpretation, along with an auxiliary proof path for classical corollaries. A key methodological contribution is the quantitative control of derivative behavior via refined Bernstein-approximation estimates (Theorem cotasi), enabling precise interpolation and approximation with controlled target-space containment. Altogether, the results yield constructive tools for representing semialgebraic-connected-by-analytic-path sets via Nash and polynomial paths and provide degree bounds in the PL case, with implications for curve selection and space-approximation problems in real algebraic geometry.
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. 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 ${\mathcal S}\subset\R^n$ be a semialgebraic set, let $x_1,\ldots,x_r\in{\mathcal S}$ be $r$ points (that we call `control points') and $0=:t_1<\ldots<t_r:=1$ be $r$ values (that we call `control times'). A natural `logistic' question concerns the existence of a smooth and semialgebraic (Nash) path $α:[0,1]\to{\mathcal S}$ that passes through the control points at the control times, that is, $α(t_k)=x_k$ for $k=1,\ldots,r$. The necessary and sufficient condition to guarantee the existence of $α$ when the number of control points is large enough and they are in general position is that $\Ss$ 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 $\Ss\subset\R^n$ has in addition dimension $n$. A related problem concerns the approximation by a Nash path of an existing continuous semialgebraic path $β:[0,1]\to{\mathcal S}$ with control points $x_1,\ldots,x_r\in\Ss$ and control times $0=:t_1<\ldots<t_r:=1$. A sufficient condition is that the (finite) set of values $η(β)$ at which $β$ is not smooth is contained in the set of regular points of ${\mathcal S}$ and $η(β)$ does not meet the set of control times.