A polynomial method over formal power series
Abstract
We give new upper bounds on families of integer vectors with prescribed coordinate multiplicities and a forbidden scalar product. Our method uses coefficient functions over formal power series and recovers the Frankl--Wilson bound for binary vectors. For vectors in {−1,0,1}n, we obtain an exponential improvement over the ordinary polynomial count in a wide range; in certain regimes, we determine the lower limit of the exponential base. We use the new bounds to make progress on several questions in combinatorial geometry. In particular, we improve the base of the exponent in Borsuk's partition problem: we construct finite subsets of ℝN requiring at least (1.37+o(1))√N parts of smaller diameter for all sufficiently large N, and obtain a base of $1.3839$ along a subsequence of dimensions.
Related
- Substitution groups of formal power series
- Dimensions of a ring and its formal power series ring
- Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings
- Finitistic dimensions of formal power series rings
- Coherence and the Hilbert-syzygy identity for power series over valuation domains
BibTeX
@misc{akhiiarov2026polynomial,
title = {{A polynomial method over formal power series}},
author = {Artur Akhiiarov and Mikhail Koshelev and Andrey Kupavskii and Andrei Raigorodskii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39460},
eprint = {2610.39460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39460},
}
RIS
TY - UNPB
TI - A polynomial method over formal power series
AU - Artur Akhiiarov
AU - Mikhail Koshelev
AU - Andrey Kupavskii
AU - Andrei Raigorodskii
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39460
UR - https://arxiv.org/abs/2610.39460
AB - We give new upper bounds on families of integer vectors with prescribed coordinate multiplicities and a forbidden scalar product. Our method uses coefficient functions over formal power series and recovers the Frankl--Wilson bound for binary vectors. For vectors in {−1,0,1}^n, we obtain an exponential improvement over the ordinary polynomial count in a wide range; in certain regimes, we determine the lower limit of the exponential base. We use the new bounds to make progress on several questions in combinatorial geometry. In particular, we improve the base of the exponent in Borsuk's partition problem: we construct finite subsets of ℝ^N requiring at least (1.37+o(1))^√N parts of smaller diameter for all sufficiently large N, and obtain a base of $1.3839$ along a subsequence of dimensions.
ER -