Surjective Nash maps between semialgebraic sets

TL;DR

This work addresses when a semialgebraic set $S$ admits a surjective Nash map onto another semialgebraic set $T$, by decomposing $S$ into irreducible components $S_i^*$ and studying their analytic-path connected targets $T_i$. The authors establish a core criterion: if $T=igcup_i T_i$ with each $T_i$ connected by analytic paths and $ ext{dim}(T_i)oldsymbol{ ceil} ext{dim}(S_i^*)$ for all $i$, then a surjective Nash map exists; the argument hinges on two pillars: B"archen–Schäfchen-type results for compact cases and a noncompact counterpart. A central achievement is a complete characterization of compact semialgebraic sets $S$ that are Nash images of the closed unit ball $ar{ ext{B}}_m$, requiring $S$ to be compact, connected by analytic paths, and satisfy $ ext{dim}(S) onumberoldsymbol{ le m}$. Consequences include that pure dimensional compact irreducible arc-symmetric sets are Nash images of $ar{ ext{B}}_d$, and every compact semialgebraic set is a projection of a nonsingular algebraic set whose components are Nash diffeomorphic to spheres; the paper also develops tools for elimination of inequalities and arc-symmetric representations using the simplicial prism and checkerboard decompositions. Overall, the results provide a constructive framework for realizing semialgebraic sets as Nash images or projections of algebraic models, with broad implications for inverse problems in real algebraic geometry. $S$-to-$T$ surjectivity criteria rely on the arrangement of components and the dimension constraints encoded by the inequalities $e_i= ext{dim}(T_i)\le d_i= ext{dim}(S_i^*)$, and the approach yields concrete representations via compact models and polyhedral tessellations, culminating in a unified treatment of arc-symmetric and eliminative tasks through explicit Nash mappings.

Abstract

In this work we study the existence of surjective Nash maps between two given semialgebraic sets ${\mathcal S}$ and ${\mathcal T}$. Some key ingredients are: the irreducible components ${\mathcal S}_i^*$ of ${\mathcal S}$ (and their intersections), the analytic-path connected components ${\mathcal T}_j$ of ${\mathcal T}$ (and their intersections) and the relations between dimensions of the semialgebraic sets ${\mathcal S}_i^*$ and ${\mathcal T}_j$. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps. The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: {\em a full characterization of the semialgebraic subsets ${\mathcal S}\subset{\mathbb R}^n$ that are the image of the closed unit ball $\overline{\mathcal B}_m$ of ${\mathbb R}^m$ centered at the origin under a Nash map $f:{\mathbb R}^m\to{\mathbb R}^n$}. The necessary and sufficient conditions that must satisfy such a semialgebraic set ${\mathcal S}$ are: {\em it is compact, connected by analytic paths and has dimension $d\leq m$}. Two remarkable consequences of the latter result are the following: (1) {\em pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension $d$ are Nash images of $\overline{\mathcal B}_d$}, and (2) {\em compact semialgebraic sets of dimension $d$ are projections of non-singular algebraic sets of dimension $d$ whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions)}.

Figures (8)

• Figure 1.1: Semialgebraic Teddy bear and semialgebraic Sheep (figures borrowed from fu6).
• Figure 2.1: Compact models to represent semialgebraic sets as their Nash images.
• Figure 2.2: The semialgebraic set ${\EuScript S}:=\{(4{\tt x}^2-{\tt y}^2)(4{\tt y}^2-{\tt x}^2)\geq0,\, {\tt y}\geq0\}\subset{\mathbb R}^2$ (figure borrowed from fu6)
• Figure 3.1: The polyhedra ${\EuScript K}_j$ (figure inspired by fu5).
• Figure 3.2: A picture of the situation.

Theorems & Definitions (57)

• Theorem 1.5: fu6
• Theorem 1.6: fu6
• Conjecture 1.8: Shiota
• Theorem 1.9: Nash images fe3
• Definition 1.10
• Theorem 1.11: fe3
• Definition 1.12
• Theorem 1.13
• Theorem 1.14: Surjective Nash maps
• Theorem 1.15: Bärchen-Schäfchen's Theorem