On The Spectral Properties of Discrete Landscape Functions
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 ℚ and obtain the bound [ℚ(ck(N)):ℚ]≤φ(N) for every odd k. The Chandra--Jain conjecture, asserting that [ℚ(c1(N)):ℚ]=φ(N) for all N≥3, 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≤ N≤50 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.
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BibTeX
@misc{chandra2026spectral,
title = {{On The Spectral Properties of Discrete Landscape Functions}},
author = {Aryaman Chandra and S. R. Jain},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.11372},
eprint = {2608.11372},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11372},
}
RIS
TY - UNPB
TI - On The Spectral Properties of Discrete Landscape Functions
AU - Aryaman Chandra
AU - S. R. Jain
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.11372
UR - https://arxiv.org/abs/2608.11372
AB - 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 ℚ and obtain the bound [ℚ(c_k(N)):ℚ]≤φ(N) for every odd k. The Chandra--Jain conjecture, asserting that [ℚ(c_1(N)):ℚ]=φ(N) for all N≥3, 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≤ N≤50 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.
ER -