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| Autor principal: | |
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| Formato: | Preprint |
| Publicado em: |
2026
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| Assuntos: | |
| Acesso em linha: | https://arxiv.org/abs/2601.18831 |
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Sumário:
- We revisit the classical Unit Distance Problem posed by Erdős in 1946. While the upper bound of $O(n^{4/3})$ established by Spencer, Szemer'edi, and Trotter (1984) is tight for systems of pseudo-circles, it fails to account for the algebraic rigidity inherent to the Euclidean metric. By integrating structural rigidity decomposition with the theory of Cayley-Menger varieties, we demonstrate that unit distance graphs exceeding a critical density must contain rigid bipartite subgraphs. We prove a "Flatness Lemma," supported by symbolic computation of the elimination ideal, showing that the configuration variety of a unit-distance $K_{3,3}$ (and by extension $K_{4,4}$) in $\mathbb{R}^2$ is algebraically singular and collapses to a lower-dimensional locus. This dimensional reduction precludes the existence of the amorphous, high-incidence structures required to sustain the $n^{4/3}$ scaling, effectively improving the upper bound for non-degenerate Euclidean configurations.