An Explicit Polynomial Counterexample to Connes' Embedding Conjecture
Abstract
We construct an explicit Hermitian polynomial in six selfadjoint variables, f=−1+ω+ω*+M∑j=115(2−uj−uj*), with integer coefficients, degree $72$, and exactly $33$ monomials. Here ω,u1,…,u15 are specified words, * reverses words, and M is a specified positive integer. Its normalized trace is at least 3/4 on selfadjoint matrix contraction tuples of every dimension, but equals −1 at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus f is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in ℤ[i]=ℤ+iℤ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.
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BibTeX
@misc{wang2026explicit,
title = {{An Explicit Polynomial Counterexample to Connes' Embedding Conjecture}},
author = {Jiaqi Wang and Lihong Zhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01536},
eprint = {2610.01536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01536},
}
RIS
TY - UNPB
TI - An Explicit Polynomial Counterexample to Connes' Embedding Conjecture
AU - Jiaqi Wang
AU - Lihong Zhi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.01536
UR - https://arxiv.org/abs/2610.01536
AB - We construct an explicit Hermitian polynomial in six selfadjoint variables, f=−1+ω+ω^*+M∑_j=1^15(2−u_j−u_j^*), with integer coefficients, degree $72$, and exactly $33$ monomials. Here ω,u_1,…,u_15 are specified words, * reverses words, and M is a specified positive integer. Its normalized trace is at least 3/4 on selfadjoint matrix contraction tuples of every dimension, but equals −1 at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus f is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in ℤ[i]=ℤ+iℤ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.
ER -