Quantitative Reduction Theory and Unlikely Intersections

@article{Daw2021QuantitativeRT,
  title={Quantitative Reduction Theory and Unlikely Intersections},
  author={Christopher Daw and Martin Orr},
  journal={International Mathematics Research Notices},
  year={2021}
}
We prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila… 
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