Approximation of maps from algebraic polyhedra to real algebraic varieties

2024/10/30 by Marcin Bilski, Wojciech Kucharz, Bilski, Marcin +1
Computer Science · Engineering · #14P05 #26C15 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2410.23457

openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a finite simplicial complex K in ℝⁿ and a real algebraic variety Y, by a K-regular map |K|→ Y we mean a continuous map whose restriction to every simplex in K is a regular map. A simplified version of our main result says that if Y is a uniformly retract rational variety and if k, l are integers satisfying 0≤ l≤ k, then every C^l map |K|→ Y can be approximated in the C^l topology by K-regular maps of class C^k. By definition, Y is uniformly retract rational if for every point y∈ Y there is a Zariski open neighborhood V⊂ Y of y such that the identity map of V is the composite of regular maps V→ W→ V, where W⊂ℝ^p is a Zariski open set for some p depending on y.

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