On the geometry of polynomial mappings at infinity

Anna Valette[1]; Guillaume Valette[2]

  • [1] Instytut Matematyki Uniwersytetu Jagiellońskiego, ul. S Łojasiewicza, Kraków, Poland
  • [2] Instytut Matematyczny PAN, ul. Św. Tomasza 30, 31-027 Kraków, Poland

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 2147-2163
  • ISSN: 0373-0956

Abstract

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We associate to a given polynomial map from ℂ 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

How to cite

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Valette, Anna, and Valette, Guillaume. "On the geometry of polynomial mappings at infinity." Annales de l’institut Fourier 64.5 (2014): 2147-2163. <http://eudml.org/doc/275593>.

@article{Valette2014,
abstract = {We associate to a given polynomial map from $\mathbb\{C\}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.},
affiliation = {Instytut Matematyki Uniwersytetu Jagiellońskiego, ul. S Łojasiewicza, Kraków, Poland; Instytut Matematyczny PAN, ul. Św. Tomasza 30, 31-027 Kraków, Poland},
author = {Valette, Anna, Valette, Guillaume},
journal = {Annales de l’institut Fourier},
keywords = {complex polynomial mappings; singularities at infinity; asymptotical values; intersection homology; Jacobian conjecture},
language = {eng},
number = {5},
pages = {2147-2163},
publisher = {Association des Annales de l’institut Fourier},
title = {On the geometry of polynomial mappings at infinity},
url = {http://eudml.org/doc/275593},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Valette, Anna
AU - Valette, Guillaume
TI - On the geometry of polynomial mappings at infinity
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 2147
EP - 2163
AB - We associate to a given polynomial map from $\mathbb{C}^2$ to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.
LA - eng
KW - complex polynomial mappings; singularities at infinity; asymptotical values; intersection homology; Jacobian conjecture
UR - http://eudml.org/doc/275593
ER -

References

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  2. M. Goresky, R. MacPherson, Intersection homology theory, Topology 19 (1980), 135-162 Zbl0448.55004MR572580
  3. M. Goresky, R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983), 77-129 Zbl0529.55007MR696691
  4. R. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302 Zbl0465.14012MR564475
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  6. Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), 1-35 Zbl0946.14039MR1717542
  7. Z. Jelonek, Geometry of real polynomial mappings, Math. Z. 239 (2002), 321-333 Zbl0997.14017MR1888227
  8. O. H. Keller, Ganze Cremonatransformationen Monatschr, Math. Phys. 47 (1939), 229-306 Zbl0021.15303
  9. F. Kirwan, J. Woolf, An Introduction to Intersection Homology Theory, (2006), Chapman & Hall/CRC Zbl1106.55001MR2207421
  10. T. Mostowski, Some properties of the ring of Nash functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 245-266 Zbl0335.14001MR412180
  11. G. Valette, L ∞ homology is an intersection homology, Adv. in Math. 231 (2012), 1818-1842 Zbl1258.14024MR2964625

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