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

Authors: Sĩ Tiệp Đinh, Zbigniew Jelonek and Tiến Sơn Phạm
Journal: Proc. Amer. Math. Soc. 150 (2022), 3663-3673
MSC (2020): Primary 14P10, 58A35; Secondary 14P15, 32C05, 58A07
Published electronically: May 20, 2022
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Abstract: Given a closed semi-algebraic set $X \subset \mathbb {R}^n$ and a continuous semi-algebraic mapping $G \colon X \to \mathbb {R}^m$, it will be shown that there exists an open dense semi-algebraic subset $\mathscr {U}$ of $L(\mathbb {R}^n, \mathbb {R}^m)$, the space of all linear mappings from $\mathbb {R}^n$ to $\mathbb {R}^m$, such that for all $F \in \mathscr {U}$, the image $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb {R}^m$. To do this, we study the tangent cone at infinity $C_\infty X$ and the set $E_\infty X \subset C_\infty X$ of (unit) exceptional directions at infinity of $X$. Specifically we show that the set $E_\infty X$ is nowhere dense in $C_\infty X \cap \mathbb {S}^{n - 1}$.

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Additional Information

Sĩ Tiệp Đinh
Affiliation: Institute of Mathematics, VAST, 18, Hoang Quoc Viet Road, Cau Giay District 10307, Hanoi, Vietnam
ORCID: 0000-0001-9116-4534

Zbigniew Jelonek
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
MR Author ID: 241045
ORCID: 0000-0002-1065-8688

Tiến Sơn Phạm
Affiliation: Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam

Keywords: Closedness, tangent cones at infinity, semi-algebraic sets/mappings, linear mappings, stability
Received by editor(s): April 4, 2021
Received by editor(s) in revised form: August 27, 2021
Published electronically: May 20, 2022
Additional Notes: The second author was partially supported by the grant of Narodowe Centrum Nauki number 2019/33/B/ST1/00755.
The first and the third authors were partially supported by the Vietnam Academy of Science and Technology under Grant Number ĐLTE00.01/21-22
Communicated by: Adrian Ioana
Article copyright: © Copyright 2022 American Mathematical Society