bibkey: carbone2026nash
authors: Antonio Carbone, José F. Fernando
year: 2026
title: Nash approximation of differentiable semialgebraic maps
doi: null
url: https://arxiv.org/abs/2601.13164v1
claim: Theorem 1.1 approximates differentiable semialgebraic maps from locally compact semialgebraic sets to Nash manifolds with corners by Nash maps in the strong differentiable topology.
strata_touched: []
license: citation-only
triage: anchor

Nash approximation on a compact parameter interval

The cited version is arXiv:2601.13164v1, posted 19 January 2026. Theorem 1.1 assumes a locally compact semialgebraic source S, a Nash manifold with corners Q, and an S^mu map from S to Q. It supplies Nash maps into the interior of Q arbitrarily close in the S^mu topology. The paper defines maps on a semialgebraic set through extensions to open neighborhoods; section 2.2 describes the topology using positive continuous semialgebraic tolerances for derivatives through order mu.

Quantum-volume section 218 uses only a closed compact interval, Euclidean target and mu = 2. Its constructed section extends as a C4 semialgebraic map to a neighborhood of the interval, so these hypotheses apply. The source supplies approximation, without asserting preservation of the quantum moment constraint or the prescribed sharp two-jets. Finite Hermite correction, normalization and a uniform scalar analytic implicit correction establish those additional properties in section 218. The paper does not establish the channel-specific five-dimensional three-point attainment result or a global trivialization of its fibers.