On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems
Abstract
In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity > 2 of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.
Related
- The metric Zariski multiplicity conjecture in dimension 4
- Multiplicity and solution construction in linear elliptic problems with nonlocal nonlinearities
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- Mixed local-nonlocal eigenvalue problems in the Heisenberg group: spectral theory and singular multiplicity
- Multiplicity of weak solutions for mixed variable exponent triharmonic problems with hardy potentials
BibTeX
@misc{sampaio2026arnold,
title = {{On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems}},
author = {José Edson Sampaio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19357},
eprint = {2610.19357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19357},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems
AU - José Edson Sampaio
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19357
UR - https://arxiv.org/abs/2610.19357
AB - In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity > 2 of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.
ER -