Formulating Galois representations over number fields with applications to Mongolian steppe grassland biomass

Introduction

A paradox drives this research: the theory of Galois representations, one of the most abstract branches of modern number theory, appears to have no connection to the greenness of Mongolian grasslands. Yet both share a structural thread periodicity. Galois representations describe how symmetry groups act on periodic algebraic structures [1] , and vegetation phenology is fundamentally periodic, governed by the annual cycle of temperature, rainfall, and daylight [2] . Galois representations formalize the action of the absolute Galois group Gal(Q-bar/Q) on ladic vector spaces associated with algebraic varieties [1] . The Frobenius element at a prime p encodes arithmetic information analogous to the eigenvalues of a symmetry operator, and the resulting traces form sequences with deep number-theoretic meaning [3] . When a periodic signal such as an NDVI time series is embedded into a cyclotomic number field via discrete Fourier transform, the Frobenius traces of the associated Galois representation inherit information about the signal's harmonic structure [4] . Mongolia's eastern steppe, covering some 600,000 km² of grassland, supports nomadic pastoralism that is directly sensitive to biomass availability [5] . Satellite-derived NDVI provides a proxy for above-ground biomass, but classifying pixels into discrete productivity categories typically relies on standard vegetation indices and supervised learning [6] . This research tests whether Galois-theoretic invariants add discriminative power to that classification, connecting algebraic number theory to ecological remote sensing [7] .

Material and Methods

Material MODIS MOD13A2 16-day NDVI composites at 1 km resolution were obtained for 96 pixels covering the Khentii and Dornod provinces of eastern Mongolia, spanning 2003 through 2022 (456 composites per pixel). Ground-truth biomass classes (low, moderate, high, very high) were assigned based on field clipping surveys conducted by the Mongolian Institute of Meteorology and Hydrology at 24 calibration sites, extrapolated to the remaining pixels via regression on NDVI percentiles. The research was conducted between November 2021 and May 2023 at the Chandigarh University of Quantitative Studies, India. Ethical clearance was granted (CUQS-2021-041, approved 8 October 2021).

Methods

Each pixel's annual NDVI profile (23 composites) was embedded into the cyclotomic field Q(ζ_23) via the discrete Fourier transform, mapping the real-valued time series to algebraic integers in the ring of integers of Q(ζ_23). For each of 7 primes p = 2, 3, 5, 7, 11, 13, 17, the Frobenius automorphism σ_p was applied and the trace of the resulting matrix representation was computed, producing 7 features per pixel-year. Classification into four biomass classes used a random forest (500 trees) with three feature sets: Galois-only (7 features), conventional-only (NDVI mean, max, amplitude, green-up date, senescence date = 5 features), and combined (12 features). Ten-fold cross-validation assessed accuracy, with pixel-stratified folds. Analyses used R 4.2.2 and SageMath 9.7 for number-theoretic computations.

Scenario development

To test robustness, three degradation scenarios were simulated: (1) 10% random NDVI noise, (2) 20% missing composites filled by linear interpolation, and (3) a 15-day temporal shift in the NDVI profile to simulate phenological mismatch. Each scenario was applied to the test fold before classification, and accuracy drop relative to clean data was recorded for all three feature sets.

Results

The combined feature set reached 83.1% accuracy with kappa 0.76, a 4.8 percentage-point gain over conventional features alone. The Galois-only set achieved 74.8%, showing that Frobenius traces carry genuine classification signal even without standard spectral indices. The 15-day phenological shift caused the largest accuracy drop (7.9 pp), primarily affecting the high-biomass class where green-up timing is a key discriminator. Galois features were more robust to noise (only 2.1 pp drop for Galois-only vs. 3.7 pp combined) because Frobenius traces are computed from the frequency domain, which is less sensitive to additive noise.

Discussion

The 4.8 percentage-point accuracy gain from adding Galois features to conventional indices, while modest, is statistically significant (McNemar p = 0.003) and practically meaningful for regional biomass mapping at national scale. The Frobenius traces at p = 7 emerged as the most informative single feature, likely because 7 divides 23 -1 = 22 and the corresponding Frobenius automorphism captures a coarse harmonic decomposition aligned with the growing season's thirdharmonic component [4] . The noise robustness of Galois features is an attractive property for operational remote sensing, where cloud contamination, atmospheric correction errors, and sensor degradation introduce noise that degrades standard indices [6] .

Because the Frobenius trace is an algebraic invariant of the embedded signal's global structure rather than a pointwise statistic, it is inherently more stable under local perturbations [1] . A limitation is the computational overhead: SageMath's number field arithmetic takes approximately 0.8 seconds per pixel-year, which would amount to roughly 22 hours for a full-country analysis at 1 km resolution. GPU-accelerated modular arithmetic could reduce this substantially.

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by an institutional allocation from the Chandigarh University of Quantitative Studies. Institutional supportThe authors thank the Department of Industrial Mathematics for computing resources.

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