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Computing Min-Convex Hulls in the Affine Building of \(\hbox {SL}_d\)

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Abstract

We describe an algorithm for computing the min-convex hull of a finite collection of points in the affine building of \(\hbox {SL}_d(K)\), for K a field with discrete valuation. These min-convex hulls describe the relations among a finite collection of invertible matrices over K. As a consequence, we bound the dimension of the tropical projective space needed to realize the min-convex hull as a tropical polytope.

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Acknowledgements

The author would like to thank the Max Planck Institute for Mathematics in the Sciences for its hospitality while working on this project. He was partially supported by a National Science Foundation Graduate Research Fellowship. The author is grateful to Jacinta Torres, Lara Bossinger, and Madeline Brandt for reading early drafts of this manuscript. He would also like to thank Michael Joswig and Lars Kastner for generous help on writing a polymake extension, Petra Schwer for suggesting a geometric interpretation of Lemma 3.8, and Bernd Sturmfels for much valuable discussion and feedback. Finally, he is grateful to the anonymous reviewers for their thoughtful feedback and many helpful suggestions.

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  1. Evans Hall, Berkeley, CA, USA, 94720

    Leon Zhang

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Correspondence to Leon Zhang.

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Zhang, L. Computing Min-Convex Hulls in the Affine Building of \(\hbox {SL}_d\). Discrete Comput Geom 65, 1314–1336 (2021). https://doi.org/10.1007/s00454-020-00223-x

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  • DOI: https://doi.org/10.1007/s00454-020-00223-x

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