Globally subanalytic arc-symmetric sets

TL;DR

The paper develops a framework for globally subanalytic arc-symmetric sets, introducing the $\mathscr{AR}$ topology and the C-semianalytic subclass to analyze zero-locus representations. It proves that every $\mathscr{AR}$-C-semianalytic arc-symmetric set is the zero locus of a globally subanalytic arc-analytic function, and that Nash globally subanalytic arc-symmetric sets arise as zeros of continuous globally subanalytic functions arc-analytic outside a simple normal crossings divisor. The authors deploy desingularization, blow-ups, and grid-based constructions to obtain global zero-locus representations, establishing a subanalytic analogue of arc-symmetric zero-set theory. They also show that Nash globally subanalytic arc-symmetric sets are semianalytic and propose a conjecture that all arc-symmetric globally subanalytic sets are Nash, hinting at a unifying structure for subanalytic arc-symmetric geometry with potential implications for o-minimality and real-analytic geometry.

Abstract

It is shown that every C-semianalytic arc-symmetric set can be realized as the zero locus of an arc-analytic function. As a consequence, a Nash globally subanalytic arc-symmetric set is the zero locus of a continuous globally-subanalytic function which is arc-analytic outside a simple normal crossings divisor.

Theorems & Definitions (41)

• Remark 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['thm:AR-topology']}
• Theorem 3.1
• proof
• Proposition 3.2