# 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}
}
• Published 13 November 2019
• Mathematics
• International Mathematics Research Notices
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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