On the Euler characteristic of fibres of real polynomial maps

Adam Parusiński; Zbigniew Szafraniec

Banach Center Publications (1998)

  • Volume: 44, Issue: 1, page 175-182
  • ISSN: 0137-6934

Abstract

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Let Y be a real algebraic subset of m and F : Y → n be a polynomial map. We show that there exist real polynomial functions g 1 , . . . , g s on n such that the Euler characteristic of fibres of F is the sum of signs of g i .

How to cite

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Parusiński, Adam, and Szafraniec, Zbigniew. "On the Euler characteristic of fibres of real polynomial maps." Banach Center Publications 44.1 (1998): 175-182. <http://eudml.org/doc/208880>.

@article{Parusiński1998,
abstract = {Let Y be a real algebraic subset of $^m$ and $F:Y → ^n$ be a polynomial map. We show that there exist real polynomial functions $g_1, ..., g_s$ on $^n$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_i$.},
author = {Parusiński, Adam, Szafraniec, Zbigniew},
journal = {Banach Center Publications},
keywords = {real algebraic subset; Euler characteristic},
language = {eng},
number = {1},
pages = {175-182},
title = {On the Euler characteristic of fibres of real polynomial maps},
url = {http://eudml.org/doc/208880},
volume = {44},
year = {1998},
}

TY - JOUR
AU - Parusiński, Adam
AU - Szafraniec, Zbigniew
TI - On the Euler characteristic of fibres of real polynomial maps
JO - Banach Center Publications
PY - 1998
VL - 44
IS - 1
SP - 175
EP - 182
AB - Let Y be a real algebraic subset of $^m$ and $F:Y → ^n$ be a polynomial map. We show that there exist real polynomial functions $g_1, ..., g_s$ on $^n$ such that the Euler characteristic of fibres of $F$ is the sum of signs of $g_i$.
LA - eng
KW - real algebraic subset; Euler characteristic
UR - http://eudml.org/doc/208880
ER -

References

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  2. [BW] E. Becker, T. Wöermann, On the trace formula for quadratic forms and some applications, in: Recent Advances in Real Algebraic Geometry and Quadratic Forms, Contemp. Math. 155, Amer. Math. Soc., Providence, 1994, 271-291. 
  3. [BR] R. Benedetti, J.-J. Risler, Real Algebraic and Semi-algebraic Sets, Actualités Math., Hermann, Paris, 1990. Zbl0694.14006
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  6. [He1] C. Hermite, Remarques sur le théorème de Sturm, C. R. Acad. Sci. Paris 36 (1853), 52-54. 
  7. [He2] C. Hermite, Sur l'extension du théorème de M. Sturm à un système d'équations simultanées, Oeuvres de Charles Hermite, Tome 3, ed. E. Picard, Edition Paris, Gauthier-Villars, 1912, 1-34. 
  8. [MP] C. McCrory, A. Parusiński, Algebraically constructible functions, Ann. Scient. École Norm. Sup. (4) 30 (1997), 527-552. Zbl0913.14018
  9. [MS] A. Mostowski, M. Stark, Elementy algebry wyższej, Państwowe Wydawnictwo Naukowe, Warszawa, 1963 (in Polish); English translation: Introduction to Higher Algebra, Internat. Series of Monographs on Pure and Appl. Math. 37, A Pergamon Press Book, New York, and Państwowe Wydawnictwo Naukowe, Warszawa, 1964. 
  10. [PS] A. Parusiński, Z. Szafraniec, Algebraically constructible functions and signs of polynomials, Manuscripta Math. 93 (1997), 443-456. Zbl0913.14019
  11. [PRS] P. Pedersen, M.-F. Roy, A. Szpirglas, Counting real zeros in the multivariate case, in: Computational Algebraic Geometry, F. Eyssette, A. Galligo (eds.), Progr. Math. 109, Birkhäuser, Boston, 1993, 203-223. Zbl0806.14042
  12. [Syl] J. J. Sylvester, On a theory of syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, Philos. Trans. Roy. Soc. London 143 (1853). 
  13. [V] O. Y. Viro, Some integral calculus based on Euler characteristic, in: Topology and Geometry-Rohlin Seminar, O. Y. Viro (ed.), Lecture Notes in Math. 1346, Springer, Berlin, 1988, 127-138. 

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