Lower bounds on the $g$-numbers of spheres without large missing faces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917417921282048 |
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| author | Novik, Isabella Zheng, Hailun |
| author_facet | Novik, Isabella Zheng, Hailun |
| contents | We establish several new lower bounds on the $g$-numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$-numbers in terms of the independence numbers of their graphs, extending a result of Chudnovsky and Nevo. As a consequence, we show that flag $(d-1)$-spheres -- and more generally, flag normal $(d-1)$-pseudomanifolds -- satisfy $g_2\geq (1/2-δ(d))f_0$, where $δ(d)$ is a function of $d$ with $δ(d)\to 0$ as $d\to \infty$. We further prove that, for simplicial $(d-1)$-spheres without large missing faces, an initial segment of the $g$-vector forms a level sequence, yielding additional inequalities among the $g$-numbers. Finally, we show that simplicial $4$-spheres without missing faces of dimension greater than two satisfy $g_2\geq \frac{2}{5}f_0 - \frac{6}{5}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_16905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lower bounds on the $g$-numbers of spheres without large missing faces Novik, Isabella Zheng, Hailun Combinatorics 13F55, 52B05, 05E40, 05E45 We establish several new lower bounds on the $g$-numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$-numbers in terms of the independence numbers of their graphs, extending a result of Chudnovsky and Nevo. As a consequence, we show that flag $(d-1)$-spheres -- and more generally, flag normal $(d-1)$-pseudomanifolds -- satisfy $g_2\geq (1/2-δ(d))f_0$, where $δ(d)$ is a function of $d$ with $δ(d)\to 0$ as $d\to \infty$. We further prove that, for simplicial $(d-1)$-spheres without large missing faces, an initial segment of the $g$-vector forms a level sequence, yielding additional inequalities among the $g$-numbers. Finally, we show that simplicial $4$-spheres without missing faces of dimension greater than two satisfy $g_2\geq \frac{2}{5}f_0 - \frac{6}{5}$. |
| title | Lower bounds on the $g$-numbers of spheres without large missing faces |
| topic | Combinatorics 13F55, 52B05, 05E40, 05E45 |
| url | https://arxiv.org/abs/2604.16905 |