On the monodromy action for $f(x,y)=g(x)+h(y)$

Daniel López Garcia; Fabricio Valencia

On the monodromy action for $f(x,y)=g(x)+h(y)$

Complex Variables 2024-10-02 v2 Algebraic Topology

Authors: Daniel López Garcia , Fabricio Valencia

Abstract

We solve the problem of determining under which conditions the monodromy of a vanishing cycle generates the whole homology of a regular fiber for a polynomial $f(x,y)=g(x)+h(y)$ where $h$ and $g$ are polynomials with real coefficients having real critical points and satisfying $deg(g)\cdot deg(h)\leq 2500$ and $\gcd(deg(g),deg(h))\leq 2$ . As an application, under the same assumptions we solve the tangential-center problem which is known to be linked to the weak 16th Hilbert problem.

Keywords

monodromy vanishing cycle polynomial homology tangential center weak 16th hilbert problem

Cite

@article{arxiv.2311.05563,
  title  = {On the monodromy action for $f(x,y)=g(x)+h(y)$},
  author = {Daniel López Garcia and Fabricio Valencia},
  journal= {arXiv preprint arXiv:2311.05563},
  year   = {2024}
}

Comments

11 pages. Final version accepted for publication in Proceedings of the American Mathematical Society

On the monodromy action for $f(x,y)=g(x)+h(y)$

Abstract

Keywords

Cite

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