Additive decompositions of multiplicative subgroups via differential identities
Abstract
We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson--Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of H=A+B, H∪{0}=A−A, H=Amathbin+̂ A, H∪{0}=Amathbin+̂ A, where H is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions H=A+B, building on the approach introduced by Hanson--Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev--Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on H∪{0}=A−A. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.
Related
- Commutator identities, Lie product identities, and Multiplicative Lie algebras
- Combinatorial decompositions of strict partitions and multi-sum Rogers-Ramanujan type identities
- Decompositions of ZMC Graphs and Euler-Ramanujan type identities
- Multiplicative subgroups are not restricted sumsets
- Covering finite fields by translates of multiplicative subgroups
BibTeX
@misc{cochrane2026additive,
title = {{Additive decompositions of multiplicative subgroups via differential identities}},
author = {Albert Cochrane and Chi Hoi Yip and Semin Yoo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02568},
eprint = {2608.02568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02568},
}
RIS
TY - UNPB
TI - Additive decompositions of multiplicative subgroups via differential identities
AU - Albert Cochrane
AU - Chi Hoi Yip
AU - Semin Yoo
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2608.02568
UR - https://arxiv.org/abs/2608.02568
AB - We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson--Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of H=A+B, H∪{0}=A−A, H=Amathbin+̂ A, H∪{0}=Amathbin+̂ A, where H is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions H=A+B, building on the approach introduced by Hanson--Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev--Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on H∪{0}=A−A. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.
ER -