PreprintCC BY 4.0arXiv
High dimensional Riemann--Roch spaces in linear spaces with small squares
Abstract
Let F be a function field over an algebraically closed field K and S a finite dimensional K-subspace of F. The square of S is spanned by all products of pairs of elements in S. We conjecture that if dim S2 ≤ 3 dim S − 4, then S2 must contain a Riemann--Roch space of dimension at least 2 dim S − 1 + g, where g is the genus of F. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that S is contained in a Riemann--Roch space of dimension at most 3/2dim S + 1. For the proof we study the annihilator of S and introduce the notion of weight for linear forms.
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BibTeX
@misc{tanısalı2026high,
title = {{High dimensional Riemann--Roch spaces in linear spaces with small squares}},
author = {Nihan Tanısalı and Mieke Wessel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06822},
eprint = {2609.06822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06822},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High dimensional Riemann--Roch spaces in linear spaces with small squares
AU - Nihan Tanısalı
AU - Mieke Wessel
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06822
UR - https://arxiv.org/abs/2609.06822
AB - Let F be a function field over an algebraically closed field K and S a finite dimensional K-subspace of F. The square of S is spanned by all products of pairs of elements in S. We conjecture that if dim S^2 ≤ 3 dim S − 4, then S^2 must contain a Riemann--Roch space of dimension at least 2 dim S − 1 + g, where g is the genus of F. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that S is contained in a Riemann--Roch space of dimension at most 3/2dim S + 1. For the proof we study the annihilator of S and introduce the notion of weight for linear forms.
ER -