Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
See all formats and pricing
More options …
Volume 30, Issue 6

Issues

Compactness criteria for real algebraic sets and Newton polyhedra

Phu Phat Pham / Tien Son PhamORCID iD: http://orcid.org/0000-0003-3368-304X
  • Corresponding author
  • Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University; and Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
  • orcid.org/0000-0003-3368-304X
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-06-30 | DOI: https://doi.org/10.1515/forum-2017-0105

Abstract

Let f:n be a polynomial and 𝒵(f) its zero set. In this paper, in terms of the so-called Newton polyhedron of f, we present a necessary criterion and a sufficient condition for the compactness of 𝒵(f). From this we derive necessary and sufficient criteria for the stable compactness of 𝒵(f).

Keywords: Compact; Newton polyhedra; polynomials; real algebraic sets; stable compactness

MSC 2010: 14P25

References

  • [1]

    N. T. N. Búi and T. S. Pham, On the subanalytically topological types of function germs, Houston J. Math. 42 (2016), no. 4, 1111–1126. Google Scholar

  • [2]

    M. J. de la Puente, Real plane algebraic curves, Expo. Math. 20 (2002), no. 4, 291–314. CrossrefGoogle Scholar

  • [3]

    K. Fukuda, Frequently asked questions in polyhedral computation, preprint (2004), http://www.cs.mcgill.ca/~fukuda/soft/polyfaq/polyfaq.html.

  • [4]

    K. Fukuda, T. M. Liebling and F. Margot, Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron, Comput. Geom. 8 (1997), no. 1, 1–12. CrossrefGoogle Scholar

  • [5]

    K. Fukuda and V. Rosta, Combinatorial face enumeration in convex polytopes, Comput. Geom. 4 (1994), no. 4, 191–198. CrossrefGoogle Scholar

  • [6]

    S. G. Gindikin, Energy estimates connected with the Newton polyhedron (in Russian), Trudy Moskov. Mat. Obšč. 31 (1974), 189–236; translation in Trans. Moscow Math. Soc. 31 (1974), 193–246. Google Scholar

  • [7]

    H.-V. Hà and T.-S. Phạm, Genericity in Polynomial Optimization, Ser. Optim. Appl. 3, World Scientific, Hackensack, 2017. Google Scholar

  • [8]

    A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31. CrossrefGoogle Scholar

  • [9]

    J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press Optim. Ser. 1, Imperial College Press, London, 2010. Google Scholar

  • [10]

    M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging Applications of Algebraic Geometry, IMA Vol. Math. Appl. 149, Springer, New York (2009), 157–270. Google Scholar

  • [11]

    M. Marshall, Optimization of polynomial functions, Canad. Math. Bull. 46 (2003), no. 4, 575–587. CrossrefGoogle Scholar

  • [12]

    M. Marshall, Positive Polynomials and Sums of Squares, Math. Surveys Monogr. 146, American Mathematical Society, Providence, 2008. Google Scholar

  • [13]

    V. P. Mihaĭlov, The behavior at infinity of a class of polynomials (in Russian), Trudy Mat. Inst. Steklov. 91 (1967), 59–80; translation in Proc. Steklov Inst. Math. 91 (1967), 61–82. Google Scholar

  • [14]

    J. Milnor, Singular Points of Complex Hypersurfaces, Ann. of Math. Stud. 61, Princeton University Press, Princeton, 1968. Google Scholar

  • [15]

    K. G. Murty and S. J. Chung, Segments in enumerating faces, Math. Program. 70 (1995), no. 1, 27–45. CrossrefGoogle Scholar

  • [16]

    C. Scheiderer, Positivity and sums of squares: A guide to recent results, Emerging Applications of Algebraic Geometry, IMA Vol. Math. Appl. 149, Springer, New York (2009), 271–324. Google Scholar

  • [17]

    J. Stalker, A compactness criterion for real plane algebraic curves, Forum Math. 19 (2007), no. 3, 563–570. Web of ScienceGoogle Scholar

About the article


Received: 2017-05-11

Revised: 2017-12-17

Published Online: 2018-06-30

Published in Print: 2018-11-01


The authors were partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED), Grant 101.04-2016.05. The second author wishes to thank Jean Bernard Lasserre for inviting him to visit the LAAS–CNRS (Toulouse) in April 2017, where the final version of this paper was completed; this visiting was supported by the European Research Council (ERC) through the ERC-Advanced Grant TAMING 666981.


Citation Information: Forum Mathematicum, Volume 30, Issue 6, Pages 1387–1395, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0105.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in