Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

2020/10/10 by Dinh, Si Tiep, Jelonek, Zbigniew, Pham, Tien Son
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.04952

Abstract

Given a closed semi-algebraic set X ⊂ ℝⁿ and a continuous semi-algebraic mapping G \colon X → ℝ^m, it will be shown that there exists an open dense semi-algebraic subset \mathscrU of L(ℝⁿ, ℝ^m), the space of all linear mappings from ℝⁿ to ℝ^m, such that for all F ∈ \mathscrU, the image (F + G)(X) is a closed (semi-algebraic) set in ℝ^m. To do this, we study the tangent cone at infinity C_∞ X and the set E_∞ X ⊂ C_∞ X of (unit) exceptional directions at infinity of X. Specifically we show that the set E_∞ X is nowhere dense in C_∞ X ∩ \mathbbS^{n - 1}.

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