PreprintCC BY 4.0arXiv
The number of Tverberg partitions beyond prime powers
Abstract
Sierksma's conjecture asserts that every affine map a:Δ(d+1)(r−1)⟶ℝd has at least ((r−1)!)d unordered Tverberg partitions into r parts. Until now, lower bounds growing with d were known only when r is a prime power. As one of our main results we prove that for every integer r≥3, every decomposition r=kq+s into q=pα a prime power, k≥1 and s≥0, and every affine map a, the number Tv(a) of unordered Tverberg partitions satisfies Tv(a) ≥ q/r! (q/(α+1))⌈((r−1−2s)(d+1))/2⌉.
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BibTeX
@misc{blagojevic2026number,
title = {{The number of Tverberg partitions beyond prime powers}},
author = {Matija Blagojević and Pavle V. M. Blagojević},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32086},
eprint = {2610.32086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The number of Tverberg partitions beyond prime powers
AU - Matija Blagojević
AU - Pavle V. M. Blagojević
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32086
UR - https://arxiv.org/abs/2610.32086
AB - Sierksma's conjecture asserts that every affine map a:Δ_(d+1)(r−1)⟶ℝ^d has at least ((r−1)!)^d unordered Tverberg partitions into r parts. Until now, lower bounds growing with d were known only when r is a prime power. As one of our main results we prove that for every integer r≥3, every decomposition r=kq+s into q=p^α a prime power, k≥1 and s≥0, and every affine map a, the number Tv(a) of unordered Tverberg partitions satisfies Tv(a) ≥ q/r! (q/(α+1))^⌈((r−1−2s)(d+1))/2⌉.
ER -