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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings
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by Sĩ Tiệp Đinh, Zbigniew Jelonek and Tiến Sơn Phạm PDF
Proc. Amer. Math. Soc. 150 (2022), 3663-3673 Request permission


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
  • Email:
  • 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
  • Email:
  • Tiến Sơn Phạm
  • Affiliation: Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam
  • Email:
  • 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
  • © Copyright 2022 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 150 (2022), 3663-3673
  • MSC (2020): Primary 14P10, 58A35; Secondary 14P15, 32C05, 58A07
  • DOI:
  • MathSciNet review: 4446220