AI 中文总结
本文研究离散区间上离散景观函数的谱系数，推导其精确公式，证明奇数模式系数的代数次数界，提出首系数次数的猜想，约化高次模式问题并建立与Arnold猫映射迹公式的结构类比。
AI 中文摘要
我们研究离散区间上离散景观函数的谱系数。离散景观函数具有显式闭式表达式，我们由此推导得到其谱系数的精确公式。我们证明，对每个奇数模式k，系数c_k(N)是位于Q的显式阿贝尔扩张中的代数数，并给出次数界[Q(c_k(N)):Q]≤φ(N)。对首系数c_1(N)，我们对3≤N≤30的每个N进行精确计算，结果显示其次数为φ(N)，这促使我们提出猜想：对所有N≥3，[Q(c_1(N)):Q]=φ(N)。我们将更高奇数模式的次数问题约化为互素情形，并明确了额外二次因子的作用。最后，我们在景观系数的奇偶性抵消与Arnold猫映射莱夫谢茨数所基于的交替迹公式之间建立了结构类比。
英文摘要
We study the landscape function on a discretized interval. The discrete landscape has an explicit closed form, and its spectral coefficients can be computed exactly. We show that these coefficients lie in explicit abelian extensions of $\mathbb{Q}$, and obtain the bound $[\mathbb{Q}(c_k(N)):\mathbb{Q}]\leqφ(N)$ for every odd $k$. For the first coefficient, exact computation for $3\leq N\leq 30$ gives the full degree $φ(N)$ in every case, motivating the Chandra--Jain conjecture that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=φ(N)$ for all $N\geq 3$. We then reduce the higher modes to the coprime case and discuss the remaining degree question. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers. The Chandra--Jain conjecture has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814).
Commentsv2: Named conjecture in abstract; added note on proof of Conjecture 4.5 by Q. Zhou (Zenodo:10.5281/zenodo.21935814). 11 pages, 1 figure
