@book{Kaplansky1972,
 author={Kaplansky, Irving}, title={Fields and Rings},
 series={Chicago Lectures in Mathematics},
 publisher={University of Chicago Press}, address={Chicago},
 edition={Second}, year={1972}
}
@article{AraOMearaPerera2002,
 author={Ara, Pere and O'Meara, Kevin C. and Perera, Francesc},
 title={Stable finiteness of group rings in arbitrary characteristic},
 journal={Advances in Mathematics}, volume={170}, number={2},
 pages={224--238}, year={2002}, doi={10.1006/aima.2002.2075}
}
@article{ElekSzabo2004,
 author={Elek, G{\'a}bor and Szab{\'o}, Endre},
 title={Sofic groups and direct finiteness},
 journal={Journal of Algebra}, volume={280}, number={2},
 pages={426--434}, year={2004}, doi={10.1016/j.jalgebra.2004.06.023}
}
@inproceedings{Haviv2018Theta,
 author={Haviv, Ishay}, title={On Minrank and the {Lov{\'a}sz} Theta Function},
 booktitle={Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
 series={Leibniz International Proceedings in Informatics}, volume={116},
 pages={13:1--13:15}, publisher={Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
 year={2018}, doi={10.4230/LIPIcs.APPROX-RANDOM.2018.13}
}
@article{GolovnevHaviv2022,
 author={Golovnev, Alexander and Haviv, Ishay},
 title={The (Generalized) Orthogonality Dimension of (Generalized) {Kneser} Graphs: Bounds and Applications},
 journal={Theory of Computing}, volume={18}, number={22},
 pages={1--22}, year={2022}, doi={10.4086/toc.2022.v018a022}
}
@article{AlonBabaiSuzuki1991,
 author={Alon, Noga and Babai, L{\'a}szl{\'o} and Suzuki, Hiroshi},
 title={Multilinear polynomials and {Frankl--Ray-Chaudhuri--Wilson} type intersection theorems},
 journal={Journal of Combinatorial Theory, Series A}, volume={58},
 number={2}, pages={165--180}, year={1991}, doi={10.1016/0097-3165(91)90058-O}
}
@article{BargZemor2022,
 author={Barg, Alexander and Z{\'e}mor, Gilles},
 title={High-Rate Storage Codes on Triangle-Free Graphs},
 journal={IEEE Transactions on Information Theory}, volume={68},
 number={12}, pages={7787--7797}, year={2022}, doi={10.1109/TIT.2022.3191309}
}
@article{HuangXiang2023,
 author={Huang, Hexiang and Xiang, Qing},
 title={Construction of storage codes of rate approaching one on triangle-free graphs},
 journal={Designs, Codes and Cryptography}, volume={91},
 pages={3901--3913}, year={2023}, doi={10.1007/s10623-023-01278-6}
}
@article{BargSchwartzYohananov2024,
 author={Barg, Alexander and Schwartz, Moshe and Yohananov, Lev},
 title={Storage Codes on Coset Graphs with Asymptotically Unit Rate},
 journal={Combinatorica}, volume={44}, number={6},
 pages={1193--1209}, year={2024}, doi={10.1007/s00493-024-00114-2}
}
@article{Prytula2024,
 author={Prytu{\l}a, Tomasz}, title={Graphical complexes of groups},
 journal={Journal of Group Theory}, volume={27}, number={2},
 pages={223--254}, year={2024}, doi={10.1515/jgth-2021-0118}
}
@article{Jennings1941,
 author={Jennings, S. A.},
 title={The structure of the group ring of a $p$-group over a modular field},
 journal={Transactions of the American Mathematical Society}, volume={50},
 number={1}, pages={175--185}, year={1941}, doi={10.1090/S0002-9947-1941-0004626-6}
}
@article{Schwartz1980,
 author={Schwartz, J. T.},
 title={Fast Probabilistic Algorithms for Verification of Polynomial Identities},
 journal={Journal of the ACM}, volume={27}, number={4},
 pages={701--717}, year={1980}, doi={10.1145/322217.322225}
}
@incollection{Zippel1979,
 author={Zippel, Richard}, title={Probabilistic algorithms for sparse polynomials},
 booktitle={Symbolic and Algebraic Computation}, editor={Ng, Edward W.},
 series={Lecture Notes in Computer Science}, volume={72},
 publisher={Springer}, address={Berlin, Heidelberg},
 pages={216--226}, year={1979}, doi={10.1007/3-540-09519-5_73}
}
@article{Clifford1937Announcement,
 author={Clifford, A. H.},
 title={Representations Induced in an Invariant Subgroup},
 journal={Proceedings of the National Academy of Sciences of the United States of America},
 volume={23}, number={2}, pages={89--90}, year={1937}, doi={10.1073/pnas.23.2.89}
}
@article{Kaplansky1958,
 author={Kaplansky, Irving}, title={Projective modules},
 journal={Annals of Mathematics}, series={2}, volume={68}, number={2},
 pages={372--377}, year={1958}, doi={10.2307/1970252}
}
@article{DykemaJuschenko2015,
 author={Dykema, Ken and Juschenko, Kate}, title={On stable finiteness of group rings},
 journal={Algebra and Discrete Mathematics}, volume={19}, number={1},
 pages={44--47}, year={2015},
 note={\url{https://admjournal.luguniv.edu.ua/index.php/adm/article/download/1174/663}}
}
@article{GurevichHadani2009,
 author={Gurevich, Shamgar and Hadani, Ronny},
 title={On the diagonalization of the discrete {Fourier} transform},
 journal={Applied and Computational Harmonic Analysis}, volume={27},
 pages={87--99}, year={2009}, doi={10.1016/j.acha.2008.11.003}
}
@incollection{Gottschalk1973,
 author={Gottschalk, Walter}, title={Some general dynamical notions},
 editor={Beck, Anatole}, booktitle={Recent Advances in Topological Dynamics},
 series={Lecture Notes in Mathematics}, volume={318},
 publisher={Springer}, address={Berlin, Heidelberg},
 pages={120--125}, year={1973}, doi={10.1007/BFb0061728}
}
@article{Gromov1999,
 author={Gromov, Mikhail}, title={Endomorphisms of symbolic algebraic varieties},
 journal={Journal of the European Mathematical Society}, volume={1},
 number={2}, pages={109--197}, year={1999}, doi={10.1007/PL00011162}
}
@article{Weiss2000,
 author={Weiss, Benjamin}, title={Sofic groups and dynamical systems},
 journal={Sankhy{\=a}: The Indian Journal of Statistics, Series A},
 volume={62}, number={3}, pages={350--359}, year={2000}
}
@article{CSCCoornaert2006,
 author={Ceccherini-Silberstein, Tullio and Coornaert, Michel},
 title={Linear cellular automata: {Garden} of {Eden Theorem}, {L}-surjunctivity and group rings},
 journal={Algebra and Discrete Mathematics}, note={no.~2},
 pages={22--35}, year={2006}
}
@article{Phung2023,
 author={Phung, Xuan Kien},
 title={A geometric generalization of {Kaplansky}'s direct finiteness conjecture},
 journal={Proceedings of the American Mathematical Society}, volume={151},
 number={7}, pages={2863--2871}, year={2023}, doi={10.1090/proc/16333}
}
@article{CSCPhungGroup2025,
 author={Ceccherini-Silberstein, Tullio and Coornaert, Michel and Phung, Xuan Kien},
 title={First-order model theory and {Kaplansky}'s stable finiteness conjecture for surjunctive groups},
 journal={Groups, Geometry, and Dynamics}, volume={19},
 pages={495--503}, year={2025}, doi={10.4171/GGD/885}
}
@article{CSCPhungMonoid2025,
 author={Ceccherini-Silberstein, Tullio and Coornaert, Michel and Phung, Xuan Kien},
 title={Stable finiteness of monoid algebras and surjunctivity},
 journal={Theoretical Computer Science}, volume={1042},
 pages={115228}, year={2025}, doi={10.1016/j.tcs.2025.115228}
}
@misc{OpenAICharacteristicTwo2026,
 author={{OpenAI}},
 title={{A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two}},
 howpublished={OpenAI Math Release preprint
               \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026}},
 year={2026}
}
