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PreprintCC BY 4.0arXiv

On the topology of fibers of complex polynomial maps

  • Laurentiu Maxim
  • John Messina
Open at sourcePDFDOICite

Abstract

We recast known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive sharp upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tibăr, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.

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Cite

BibTeXRIS
BibTeX
@misc{maxim2026topology,
  title = {{On the topology of fibers of complex polynomial maps}},
  author = {Laurentiu Maxim and John Messina},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.04320},
  eprint = {2608.04320},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.04320},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the topology of fibers of complex polynomial maps
AU  - Laurentiu Maxim
AU  - John Messina
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2608.04320
UR  - https://arxiv.org/abs/2608.04320
AB  - We recast known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive sharp upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tibăr, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.
ER  -