AI 中文总结
该研究运用p进分析与代数数论，探讨d维离散环的硬核熵常数κ_d的代数性，收集其代数性判据并界定相关素数幂的可能值数量。
AI 中文摘要
我们运用p进分析与代数数论的工具和技术，研究硬方块熵常数及其高维类似物的代数性。具体而言，我们研究a_d(n)的算术性质，其中a_d(n)是d维离散环上独立集的数量，以及与之相关的熵常数κ_d=lim_{n→∞}a_d(n)^{1/n^d}。对于d>1，κ_d是代数数还是超越数尚不清楚。利用序列a_d(p^k)对每个素数p均p进收敛这一事实，我们收集了κ_d代数性的判据，并对满足a_d(p^k)=κ_d^{p^{kd}}的素数幂p^k的可能值数量进行了界定。
英文摘要
We use tools and techniques from $p$-adic analysis and algebraic number theory to study the algebraicity of the hard square entropy constant and its high dimensional analogues. Specifically, we study arithmetic properties of $a_d(n)$, the number of independent sets in the $d$-dimensional discrete torus, and the associated entropy constants $κ_d=\lim_{n\to\infty}a_d(n)^{1/n^d}$. It is not known whether $κ_d$ is algebraic or transcendental for $d>1$. Using the fact that the sequence $a_d(p^k)$ converges $p$-adically for every prime $p$ and other arithmetic facts, we present a collection of criteria for the algebraicity of $κ_d$ and bound the number of possible values of prime powers $p^k$ for which $a_d(p^k)=κ_d^{p^{kd}}$.
Comments33 pages, comments are welcome
