Year: 2026
Type: preprint
Abstract: 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 $k$. The Chandra--Jain conjecture, asserting that $[\mathbb{Q}(c_1(N)):\mathbb{Q}]=φ(N)$ for all $N\geq3$, has since been proved by Q. Zhou (Zenodo, doi:10.5281/zenodo.21935814). We then reduce higher modes to the coprime case. Exact computations for all 624 pairs with $3\leq N\leq50$ reveal only the degrees $φ(N/d)$ and $2φ(N/d)$ for non-coprime modes, where $d=\gcd(N,k)$. This motivates the Second Chandra--Jain Conjecture that this degree dichotomy holds generally. We also discuss connections with parity, the Arnold cat map, and Lefschetz numbers.
Source: arXiv (Cornell University)
Authors: Chandra, Aryaman, Jain, S. R.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202350565