A group-action Szemer\'edi-Trotter theorem and applications to orchard problems in all characteristics
We establish a group-action version of the Szemer\'edi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $...
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| creator | Jing, Yifan Zou, Tingxiang |
| description | We establish a group-action version of the Szemer\'edi-Trotter theorem over
any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an
Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of
collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k =
\mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by
Bays, Dobrowolski, and the second author. |
| doi_str_mv | 10.48550/arxiv.2411.13084 |
| format | Article |
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any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an
Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of
collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k =
\mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by
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any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an
Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of
collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k =
\mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by
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any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an
Elekes-Szab\'o-type application, we obtain quantitative bounds on the number of
collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k =
\mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by
Bays, Dobrowolski, and the second author.</abstract><doi>10.48550/arxiv.2411.13084</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | A group-action Szemer\'edi-Trotter theorem and applications to orchard problems in all characteristics |
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