arXiv 2602.169932026-02-19math.ACmath.CO

通过极化神经理想同调不变量的神经码

Neural codes via homological invariants of polarized neural ideals

Bryn Mawr College（布林茅尔学院）

Selvi Kara, Ellie Lew

备注：: 29 pages, 4 figures

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文通过极化神经理想同调不变量研究神经码的极值行为，证明了极值条件与立方体几何的关系，并构造了达到极值的码族。

摘要：

对于一个神经码C⊆F₂ⁿ，极化神经理想J_C的典范形式生成元会得到一个自由单项式理想P(J_C)⊆k[x₁,…,xₙ,y₁,…,yₙ]，即极化神经理想，以及一个关联的simplicial复形Δ_C，即极化复形。我们通过Δ_C的拓扑结构研究P(J_C)的分级不变量pd(P(J_C))和reg(P(J_C))，显示简单的几何特征（Hamming立方体F₂ⁿ的Hamming距离）组织其极值行为。我们证明reg(P(J_C))≤2n−1，当且仅当C由F₂ⁿ删除一个反向配对对得到。利用Δ_C诱导子复形的连通性性质，我们得到pd(P(J_C))≤2n−3，并给出一个显式码族达到等式，每个码由反向配对对组成。在另一端，我们识别出最小值背后的立方体几何：reg(P(J_C))=1迫使C为F₂ⁿ的坐标子立方体，而pd(P(J_C))=0迫使C为一个补集。最后，我们构造了实现固定n的(pd,reg)-图中大区域的码族。

英文摘要：

For a neural code $\mathcal{C}\subseteq\mathbb{F}_2^n$, polarizing the canonical form generators of the neural ideal $J_{\mathcal{C}}$ yields a squarefree monomial ideal $\mathcal{P}(J_{\mathcal{C}})\subset k[x_1,\dots,x_n,y_1,\dots,y_n]$, the polarized neural ideal, and an associated simplicial complex $Δ_{\mathcal{C}}$, the polar complex. We study the graded invariants $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))$ and $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))$ via the topology of $Δ_{\mathcal{C}}$, showing that simple geometric features of the Hamming cube $\mathbb{F}_2^n$ (with Hamming distance) organize their extremal behavior. We prove $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-1$, with equality precisely when $\mathcal{C}$ is obtained from $\mathbb{F}_2^n$ by deleting an antipodal pair. Using connectedness properties of induced subcomplexes of $Δ_{\mathcal{C}}$, we obtain $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))\le 2n-3$, and we give an explicit family of codes attaining equality, each consisting of antipodal pairs. At the opposite end, we identify the cube geometry behind the smallest values: $\operatorname{reg}(\mathcal{P}(J_{\mathcal{C}}))=1$ forces $\mathcal{C}$ to be a coordinate subcube of $\mathbb{F}_2^n$, while $\operatorname{pd}(\mathcal{P}(J_{\mathcal{C}}))=0$ forces $\mathcal{C}$ to be the complement of one. Finally, we construct families realizing large regions of the $(\operatorname{pd},\operatorname{reg})$-plot for fixed $n$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文