A note on singularities at infinity of complex polynomials

Adam Parusiński

Banach Center Publications (1997)

  • Volume: 39, Issue: 1, page 131-141
  • ISSN: 0137-6934

Abstract

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Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family f ¯ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the C ∞ -triviality of f. If the support of sheaf of vanishing cycles of f ¯ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.

How to cite

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Parusiński, Adam. "A note on singularities at infinity of complex polynomials." Banach Center Publications 39.1 (1997): 131-141. <http://eudml.org/doc/208656>.

@article{Parusiński1997,
abstract = {Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family $\overline\{f\}$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the $C^\infty $-triviality of f. If the support of sheaf of vanishing cycles of $\overline\{f\}$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.},
author = {Parusiński, Adam},
journal = {Banach Center Publications},
keywords = {bifurcation set; vanishing cycles; Malgrange condition},
language = {eng},
number = {1},
pages = {131-141},
title = {A note on singularities at infinity of complex polynomials},
url = {http://eudml.org/doc/208656},
volume = {39},
year = {1997},
}

TY - JOUR
AU - Parusiński, Adam
TI - A note on singularities at infinity of complex polynomials
JO - Banach Center Publications
PY - 1997
VL - 39
IS - 1
SP - 131
EP - 141
AB - Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family $\overline{f}$ of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the $C^\infty $-triviality of f. If the support of sheaf of vanishing cycles of $\overline{f}$ is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in this case, the generic fibre of f has the homotopy type of a bouquet of spheres.
LA - eng
KW - bifurcation set; vanishing cycles; Malgrange condition
UR - http://eudml.org/doc/208656
ER -

References

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