发表机构
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院; 中国科学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了一个显式整数系数Hermite多项式，作为Connes嵌入猜想代数表述的反例，其归一化迹在矩阵压缩上至少为3/4，在群von Neumann代数中为-1。
AI 中文摘要
我们构造了一个显式的Hermite多项式$f$，其系数为整数，次数为12，涉及65个自伴变量，其归一化迹在任意维数下对每一组自伴矩阵压缩元组至少为$3/4$，而在一个群von Neumann代数中指定的自伴酉元组上等于$-1$。因此，对于$0\le\varepsilon<1$，$f+\varepsilon$位于交换子模的压缩二次模之外，从而给出了Connes嵌入猜想代数表述的一个显式反例。结合Kun和Thom的群构造、Thom的归一化论证以及Alekseev、Liu和Thom的谱修正定理，我们确定了一个显式正整数$\mu$，使得$f=1-(P Q)^2+\mu\sum_{\nu=1}^{825}E_\nu^*E_\nu+\mu\sum_{j=1}^{65}(1-X_j^2)^2$。这里$P,Q$编码共轭对合，$E_\nu$编码关系缺陷。
英文摘要
We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+ω+ω^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $ω,u_1,\ldots,u_{15}$ 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 $\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}$ 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.
Comments46 pages. Substantially revised, with a simplified main construction, a quartic counterexample, and an improved two-variable encoding
