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arXiv · 2606.21449

Hilbert's 16th problem for arrangements of curves on a surface

Giacomo Maletto
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Abstract

We introduce a combinatorial structure $(n,W,T)$ encoding the topological type of a curve transverse to a fixed cellular arrangement of curves on a compact real surface, in terms of intersection numbers, Dyck words and rooted trees. We apply this formalism to analyze a natural generalization of Hilbert's 16th problem to arrangements of curves. We obtain a complete classification of arrangements of three lines and a cubic, and a partial classification of arrangements of three lines and a quartic. This is achieved using B\'ezout-type obstructions, Viro's patchworking and translations, and by developing the Julia library NWT to handle large databases of curves.

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Giacomo Maletto. 2026-06-19. Hilbert's 16th problem for arrangements of curves on a surface. https://arxiv.org/abs/2606.21449

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